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+Optimal Scaling Results for a Wide Class of Proximal MALA
+Algorithms
+Francesca R. Crucinio∗1, Alain Durmus2, Pablo Jim´enez3, and Gareth O. Roberts4
+1CREST, ENSAE Paris
+2Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, France, Institut
+Polytechnique de Paris
+3Sorbonne Universit´e and Universit´e Paris Cit´e, CNRS, Laboratoire de
+Probabilit´es, Statistique et Mod´elisation, F-75005 Paris, France
+4Department of Statistics, University of Warwick
+Abstract
+We consider a recently proposed class of MCMC methods which uses proximity maps in-
+stead of gradients to build proposal mechanisms which can be employed for both differentiable
+and non-differentiable targets. These methods have been shown to be stable for a wide class
+of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms
+(MALA); and have found wide application in imaging contexts. The wider stability properties
+are obtained by building the Moreau-Yoshida envelope for the target of interest, which depends
+on a parameter λ. In this work, we investigate the optimal scaling problem for this class of
+algorithms, which encompasses MALA, and provide practical guidelines for the implementation
+of these methods.
+Contents
+1
+Introduction
+2
+2
+Proximal MALA Algorithms
+5
+3
+Optimal scaling of Proximal MALA
+7
+3.1
+Differentiable targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2
+Laplace target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4
+Practical Implications and Numerical Simulations
+13
+4.1
+Numerical Experiments
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+5
+Discussion
+15
+∗Corresponding author: francesca.crucinio@gmail.com
+1
+arXiv:2301.02446v1  [stat.CO]  6 Jan 2023
+
+6
+Proof of the Result for the Laplace distribution
+16
+6.1
+Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+6.2
+Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+6.3
+Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+6.4
+Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+A Proof of Theorem 1
+38
+A.1 Auxiliary Results for the Proof of Case (a) . . . . . . . . . . . . . . . . . . . . . . . .
+39
+A.2 Auxiliary Results for the Proof of Case (b)
+. . . . . . . . . . . . . . . . . . . . . . .
+47
+A.3 Auxiliary Results for the Proof of Case (c) . . . . . . . . . . . . . . . . . . . . . . . .
+51
+A.4 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+54
+B Numerical Experiments
+54
+B.1
+Differentiable Targets
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+54
+B.2
+Laplace Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+55
+C Taylor Expansions for the Results on Differentiable Targets
+63
+C.1 Coefficients of the Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . .
+63
+C.1.1
+Case (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+63
+C.1.2
+Case (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+63
+C.1.3
+Case (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+64
+C.2
+Taylor Expansions of the Log-acceptance Ratio . . . . . . . . . . . . . . . . . . . . .
+66
+C.2.1
+R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+66
+C.2.2
+R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+67
+C.3 Derivatives of the Proximity Map for Differentiable Targets . . . . . . . . . . . . . .
+71
+D Moments and Integrals for the Laplace Distribution
+73
+D.1 Moments of Acceptance Ratio for the Laplace Distribution
+. . . . . . . . . . . . . .
+73
+D.2 Bound on Second Moment of Acceptance Ratio for the Laplace Distribution . . . . .
+79
+D.3 Additional Integrals for the Laplace Distribution . . . . . . . . . . . . . . . . . . . .
+83
+D.4 Integrals for Moment Computations
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+84
+D.4.1
+First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+84
+D.4.2
+Second Moment
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+87
+D.4.3
+Third Moment
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+88
+1
+Introduction
+Gradient-based Markov chain Monte Carlo methods have proved to be very successful at sampling
+from high-dimensional target distributions [9].
+The key to their success is that in many cases
+their mixing time appears to be better than their competitor algorithms which do not use gradient
+information (see for example [34]), while their implementation has similar computational cost.
+Indeed, gradients of target densities can often be computed with computational complexity (in
+dimension d) which scales no worse than evaluation of the target density itself.
+Gradient-based MCMC methods are mainly motivated from stochastic processes constructed
+to have the target density as limiting distribution [25, 8, 6, 44]. Our analysis will concentrate on
+2
+
+Metropolis Adjusted Langevin Algorithm (MALA) and its proximal variants which are based on
+the Langevin diffusion
+dLt = dBt + ∇ log π(Lt)
+2
+dt ,
+(1)
+where π denotes the target density with respect to the Lebesgue measure. It is well-known that un-
+der appropriate conditions, (1) defines a continuous-time Markov process associated with a Markov
+semigroup which is reversible with respect to π. From this observation, it has been suggested to
+use a Euler-Maruyama (EM) approximation of (1). This scheme has been popularized in statistics
+by [20] and referred to as the Unadjusted Langevin Algorithm (ULA) in [36]. Due to the time-
+discretization, ULA does not have π as stationary distribution. To address this problem, [39] and
+independently Besag in his contribution to [20] proposed to add a Metropolis acceptance step at
+each iteration of the EM scheme, leading to the Metropolis Adjusted Langevin Algorithm (MALA)
+following [36] who also derive basic stability analysis. The accept/reject step in this algorithm
+confers two significant advantages: it ensures that the resulting algorithm has exactly the correct
+invariant distribution, while step sizes can be chosen larger than in the unadjusted case as there
+is not need to make step size small to reduce discretization error.
+On the other hand, MALA
+algorithms are typically hard to analyze theoretically (see e.g. [7, 13, 16]). However, [34] (see also
+[5, 32]) have established that MALA has better convergence properties than the Random Walk
+Metropolis (RWM) algorithm with respect to the dimension d from an optimal scaling perspective
+(see also [33]).
+Whereas gradient-based methods have been successively applied and offer interesting features,
+they are typically less robust than their vanilla alternatives (for example see [36]) while intuition
+suggests, and existing underpinning theory requires, that target densities need to be sufficiently
+smooth for the gradients to be aiding Markov chain convergence. Moreover, while gradient-based
+MCMC have been successful for smooth densities, there is no reason to believe that they should be
+effective for densities which are not differentiable at a subset D ⊆ Rd. For non-smooth densities,
+[30] proposes modified gradient-based algorithms. Their proposed P-MALA algorithm is inspired
+by the proximal algorithms popular in the optimization literature (e.g. [29]). The main idea is
+to approximate the (possibly non differentiable but) log-concave target density π ∝ exp(−G) by
+substituting the potential G with its Moreau-Yoshida envelope Gλ (see (3) below for its definition),
+to obtain a distribution πλ whose level of smoothness is controlled by the proximal parameter λ > 0.
+Given this smooth approximation to π one can then build proposals based on time discretizations
+of the Langevin diffusion targeting πλ [30, 14]:
+ξk+1 = ξk − σ2
+2 ∇Gλ(ξk) + σZk+1 ,
+(2)
+where σ2 > 0 is a fixed stepsize and (Zk)k∈N∗ is a sequence of i.i.d. zero-mean Gaussian random
+variables with identity covariance matrix. Our aims in this paper are broadly to provide theo-
+retical underpinning for a slightly larger family of proximal MALA algorithms, analyze how these
+methods scale with dimension, and to give insights and practical guidance into how they should be
+implemented supported by the theory we establish.
+Proximal optimization and MCMC methods proved to be particularly well-suited for image
+estimation, where penalties involving the sparsity inducing norms are common [30, 14, 43]. Similar
+targets are also common in sparse regression contexts [2, 19, 46]. In these situations, the set of
+non-differentiability points for the target density π is a null set for the Lebesgue measure, and,
+3
+
+following [12], we shall focus on this case. However, in contrast to the conclusions of [12] for RWM,
+we shall demonstrate that optimal scaling of proximal MALA may be affected by non-smoothness.
+More precisely, in this work, we first extend the results of [31] and consider a wider range of
+proximal MALA algorithms, as well as a wider class of finite dimensional target distributions. We
+let both the steps size σ2 and the regularization parameter λ depend on the dimension d of the
+target and find that the scaling properties of proximal MALA depend on the relative speed at which
+λ and σ converge to 0 as d → ∞. We start by considering a class of sufficiently differentiable target
+distributions π to which MALA can also be applied, to allow direct comparison between MALA
+and proximal MALA and thus between a gradient-based method and one which approximates the
+gradient through proximal operators. When λ goes to 0 at least as fast as σ2, we find that the
+scaling properties of proximal MALA are equivalent to those of MALA (i.e. σ2 should decay as
+d���1/3; see Theorem 1–(b), Theorem 1–(c) and Theorem 2); when λ converges to 0 more slowly than
+σ2, proximal MALA is less efficient than MALA with σ2 decaying as d−1/2 (Theorem 1–(a)). As
+in some cases the proximal operator for a given distribution π is cheaper to compute than ∇ log π
+[29, 11, 30], we anticipate that proximal MALA with an appropriately tuned λ might provide a
+cheaper alternative to MALA retaining similar scaling properties.
+We then turn to the optimal scaling of proximal MALA applied to the Laplace distribution
+π(x) ∝ e−|x|. We focus on this particular non-smooth target since it is the most widely used in
+applications of proximal MALA, including image deconvolution [30, 14, 43], LASSO, and sparse
+regression [2, 19, 46]. We establish that non-differentiability of the target even at one point leads
+to a different optimal scaling than MALA. In particular, the step size has to scale as d−2/3 and not
+as d−1/3.
+This appears to be a new optimal scaling for Metropolis MCMC algorithms which is between
+the one of RWM and MALA. From the conclusion for smooth target distributions, we restrict our
+study to the choice of λ going to 0 at least as fast as σ2.
+The proof of the result for the differentiable case extends that of [34] for MALA, while the
+structure of the proof for the Laplace target is similar to that of [12] and constitutes the main
+element of novelty in this paper. As a special case of the result for the Laplace distribution, we also
+obtain the optimal scaling for MALA on Laplace targets. We point out that the strategy adopted
+in the proof of this result is not unique to the Laplace distribution, and could be applied to other
+distributions provided that the required integrals can be obtained.
+To sum up, our main contributions are:
+1) We extend the result of [31] beyond the Gaussian case, covering all finite dimensional (suffi-
+ciently) differentiable targets, and show that, in some cases, proximal MALA affords the same
+scaling properties of MALA if the proximal parameter λ is chosen appropriately.
+2) Motivated by applications in imaging and sparse regression applications, we study the scaling of
+proximal MALA methods for the Laplace target, and show that for values of λ decaying sufficiently
+fast, the optimal scaling of proximal MALA, i.e. the choice for σ2, is different from the one for
+MALA on differentiable targets and is of order d−2/3.
+3) We use the insights obtained with the aforementioned results to provide practical guidelines for
+the selection of the proximal parameter λ.
+Organization of the paper
+The paper is structured as follows. In Section 2, we rigorously
+introduce the class of proximal MALA algorithms that are studied and discuss related works on
+optimal scaling for MCMC algorithms. In Section 3.1 we state the main result for differentiable
+targets, showing that the scaling properties of proximal MALA depend on the relative speed at
+4
+
+which λ goes to 0 with respect to σ. In Section 3.2 we obtain a scaling limit for proximal MALA
+when π is a Laplace distribution, as a special case of our result we also obtain the scaling properties
+of a sub-gradient version of MALA for this target.
+We collect in Section 4 the main practical
+takeaways from these results and discuss possible extensions in Section 5. Finally, in Section 6 we
+prove the result for the Laplace distribution. The proof of the result for differentiable targets is
+postponed to Appendix A.
+2
+Proximal MALA Algorithms
+We now introduce the general class of proximal MALA algorithms, first studied in [30]. This class
+of algorithms aims at sampling from a density with respect to the Lebesgue measure on Rd of the
+form π(x) = exp(−G(x))/
+�
+Rd exp(−G(˜x))d˜x, with G satisfying the following assumption
+A0. The function G : Rd → R is convex, proper and lower semi-continuous.
+The main idea behind proximal MALA is to approximate the (possibly non differentiable) target
+density π by approximating the potential G with its Moreau-Yoshida envelope Gλ : Rd → R defined
+for λ > 0 by
+Gλ(x) = min
+u∈Rd[G(u) + ∥u − x∥2/(2λ)] .
+(3)
+Since G is supposed to be convex, by [38, Theorem 2.26], the Moreau-Yoshida envelope is well-
+defined, convex and continuously differentiable with
+∇Gλ(x) = λ−1(x − proxλ
+G(x)) ,
+proxλ
+G(x) = arg min
+u∈Rd[G(u) + ∥u − x∥2/(2λ)] .
+(4)
+The proximity operator x �→ proxλ
+G(x) behaves similarly to a gradient mapping and moves points
+in the direction of the minimizers of G. In the limit λ → 0 the quadratic penalty dominates (4)
+and the proximity operator coincides with the identity operator, i.e. proxλ
+G(x) = x; in the limit
+λ → ∞, the quadratic penalty term vanishes and (4) maps all points to the set of minimizers of G.
+It was shown in [14, Proposition 1] that, under A0,
+�
+Rd exp(−Gλ(x))dx < ∞, and therefore
+the probability density πλ ∝ exp(−Gλ) is well-defined. In addition, it has been shown that ∥π −
+πλ∥TV → 0 as λ → 0. Based on this observation and since as we have emphasized πλ is now
+continuously differentiable, it has been suggested in [30, 14] to use the discretization of the Langevin
+diffusion associated with πλ given by (2), which can be rewritten using (4) as
+ξk+1 =
+�
+1 − σ2
+2λ
+�
+ξk + σ2
+2λ proxλ
+G(ξk) + σZk+1 .
+(5)
+Similarly to other MCMC methods based on discretizations of the Langevin diffusion (e.g.
+[36]), one can build unadjusted schemes which target πλ, expecting draws from these schemes to
+be close to draws from π for small enough λ, or add a Metropolis-Hastings step to ensure that the
+resulting algorithm targets π. Unadjusted proximal MCMC methods have been analyzed in [14];
+in this paper we focus on Metropolis adjusted proximal MCMC methods and study their scaling
+properties. More precisely, at each step k and given the current state of the Markov chain Xk,
+a candidate Yk+1 is generated from the transition density associated to (5), (x, y) �→ q(x, y) =
+ϕ(y; [1−σ2/(2λ)]x+σ2 proxλ
+G(x)/2λ, σ2 Id), where ϕ(· ; u, Σ) stands for the d-dimension Gaussian
+density with mean u and covariance matrix Σ. Given Xk and Yk+1, Then, the next state is set as:
+Xk+1 = Yk+1bk+1 + Xk(1 − bk+1) , bk+1 = 1R+
+�π(Yk+1)q(Yk+1, Xk)
+π(Xk)q(Xk, Yk+1) ∧ 1 − Uk+1
+�
+,
+(6)
+5
+
+where (Ui)i∈N∗ is a sequence of i.i.d. uniform random variables on [0, 1].
+The value of λ characterizes how close the distribution πλ is to the original target π and therefore
+how good the proposal is. Small values of λ provide better approximations to π and therefore better
+proposals (see [14, Proposition 1]), while larger values of λ provide higher levels of smoothing for
+non-differentiable distributions (see [30, Figure 1]). In the case λ = σ2/2 we obtain the special case
+of proximal MALA referred to as P-MALA in [30].
+The main contribution of this paper is to analyze the optimal scaling for proximal MALA defined
+by (6).
+Optimal scaling and related works
+We briefly summarize here some examples of MCMC
+algorithms and their optimal scaling results; a full review is out of the scope of this paper and
+we only mention algorithms to which we will compare proximal MALA in the development of this
+work.
+Popular examples of Metropolis MCMC are RWM and MALA. RWM uses as a proposal the
+transition density (x, y) �→ ϕ(y ; x, σ2 Id), where σ2 > 0. The MALA scheme uses as proposal
+(x, y) �→ ϕ(y ; x + (σ2/2)∇ log π(x), σ2 Id). As we will show in Section 3.1, proximal MALA can
+be considered as an extension of MALA.
+A natural question to address when implementing Metropolis adjusted algorithms is how to set
+the parameter σ2 (variance parameter for RWM, step size parameter for MALA) to maximize the
+efficiency of the algorithm. Small values of σ2 result in higher acceptance probability and cause
+sticky behaviour, while large values of σ2 result in a high number of rejections with the chain
+(Xk)k≥0 moving slowly [35]. Optimal scaling studies aim to address this question by investigating
+how σ2 should behave with respect to the dimension d of the support of π in the high dimensional
+setting d → ∞, to obtain the best compromise.
+The standard optimal scaling set-up considers the case of d-dimensional targets πd which are
+product form, i.e.
+πd(xd) =
+d
+�
+i=1
+π(xd
+i ) ,
+(7)
+where xd
+i stands for the i-th component of xd and π is a one-dimensional probability density with
+respect to the Lebesgue measure.
+Under appropriate assumptions on the regularity of π, and
+assuming that the MCMC algorithm is initialized at stationarity, the optimal value of σ2 scales as
+ℓ/dα with ℓ > 0, α = 1 for RWM [33] and α = 1/3 for MALA [34].
+By setting α to these values, it is then possible to show that each as d → ∞ each 1-dimensional
+component of the Markov chain defined by RWM and MALA, appropriately rescaled in time,
+converges to the Langevin diffusion
+dLt = h(ℓ)1/2dBt − h(ℓ)
+2 [log π]′(x)dt ,
+where (Bt)t≥0 is a standard Brownian motion and h(ℓ), referred to as speed function of the diffusion,
+is a function of the parameter ℓ > 0 that we may tune. Indeed, it is well-known that (Lh(ℓ)t)t≥0 is
+a solution of the Langevin diffusion (1). As a result, we may identify the values of ℓ maximizing
+h(ℓ) for the algorithms at hand to approximate the fastest version of the Langevin diffusion. The
+optimal values for ℓ results in an optimal average acceptance probability of 0.234 for RWM and
+0.574 for MALA.
+6
+
+The scaling properties allow to get an intuition of the efficiency of the corresponding algorithms:
+RWM requires O(d) steps to achieve convergence on a d-dimensional target, i.e. its efficiency is
+O(d−1), while MALA has efficiency O(d−1/3). While these results are asymptotic in d, the insights
+obtained by considering the limit case d → ∞ prove to be useful in practice [35].
+In the context of non-smooth and even discontinuous target distributions, studying the simpler
+RWM algorithm applied to a class of distributions on compact intervals, [27, 28] show that the lack
+of smoothness effects the optimal scaling of RWM with respect to dimension d. More precisely,
+they show that for a class of discontinuous densities which includes the uniform distribution on
+[0, 1], the optimal scaling of RWM is of order O(d−2). On the other hand, in the case where the set
+of non-differentiability D of π is a null set with respect to the Lebesgue measure, [12] shows that
+under appropriate conditions, including Lp differentiability, the optimal scaling of RWM is of order
+O(d−1) still.
+The scaling properties of proximal MALA have been partially investigated in [31], which shows
+that P-MALA, obtained when λ = σ2/2, has the same scaling properties of MALA for the finite
+dimensional Gaussian density and for a class of infinite dimensional target measures (Theorem 2.1
+and Theorem 5.1 therein, respectively).
+3
+Optimal scaling of Proximal MALA
+We consider the same set up of [34] and briefly recalled above.
+Given a real-valued function
+g : R → R satisfying A0 we consider the i.i.d. d-dimensional target specified by (7) with
+π(x) ∝ exp(−g(x)) .
+(8)
+Since for any xd, G(xd) = �d
+i=1 g(xd
+i ), we have by [29, Section 2.1]
+proxλ
+G(xd) = (proxλ
+g(xd
+1), . . . , proxλ
+g(xd
+d))⊤ .
+It follows that the distribution of the proposal (10) with target πd in (7)-(8) is also product form
+qd(xd, yd) = �d
+i=1 q(xd
+i , yd
+i ) ,
+q(xd
+i , yd
+i ) =
+1
+(2πσ2)1/2 exp
+�
+−(yd
+i −xd
+i +σ2g′[proxλ
+g (xd
+i )]/2)
+2
+2σ2
+�
+,
+with λ > 0. For any dimension d ∈ N∗, we denote by (Xd
+k)k∈N the Markov chain defined by the
+Metropolis recursion (6) with target distribution πd and proposal density qd and associated to the
+sequence of candidate moves
+Y d
+k+1 =
+�
+1 − σ2
+2λ
+�
+Xd
+k + σ2
+2λ proxλ
+G(Xd
+k) + σZd
+k+1 .
+(9)
+As mentioned in the introduction, the focus of this work is on investigating the optimal depen-
+dence of the proposal variance σ2 on the dimension d of the target π. In this section, we make the
+dependence of the proposal variance on the dimension explicit and let σ2
+d = ℓ2/d2α and λd = c2/2d2β
+for some α, β > 0 and some constants c, ℓ independent on d. Thus, we can write λd as a function of
+σd, λd = σ2m
+d r/2, where we defined r = c2/ℓ2m > 0 and m = β/α. By writing λd as a function of
+σd we can decouple the effect of the constants c, ℓ from that of the dependence on d (i.e. α, β). The
+value of m controls the relative speed at which σd and λd converge to 0 as d → ∞, when m = 1,
+7
+
+σd and λd decay to 0 at the same rate, for m > 1 the decay of λd is faster than that of σd and for
+m < 1 the decay of λd is slower than that of σd. The parameter r allows to refine the comparison
+between σd and λd as β = α. In the limit r → 0 (i.e. λd/σd → 0 if α = β), the proposal (10)
+coincides with that of MALA, in the case m = 1, r = 1 we get the P-MALA algorithm studied
+in [30, 31]. Note that for all other values of r, m we have a family of proposals whose behaviour
+depends on r and m.
+3.1
+Differentiable targets
+We start with the case where π is continuously differentiable. Since MALA can be applied to this
+class of targets, the results obtained in this section allow direct comparison of proximal MALA
+algorithms with MALA and thus between gradient-based algorithms (MALA) and algorithms that
+use proximal operator-based approximations of the gradient (proximal MALA). If G = − log π is
+continuously differentiable, using [3, Corollary 17.6], proxλ
+G(x) = −λ∇G(proxλ
+G(x)) + x, and (5)
+reduces to
+ξk+1 = ξk − σ2
+2 ∇G(proxλ
+G(ξk)) + σZk+1 .
+(10)
+Hence, the value of λ controls how close to ξk is the point at which the gradient is evaluated.
+For λ → 0, the proximal MALA proposal becomes arbitrarily close to that of MALA, while, as λ
+increases (10) moves away from MALA.
+Our main result, Theorem 1 below, shows that the relative speed of decay (i.e. m) influences
+the optimal scaling of the resulting proximal MALA algorithm, while the constant r influences the
+speed function of the limiting diffusion.
+We make the following assumptions on the regularity of g.
+A1. g is a C8-function whose derivatives are bounded by some polynomial: there exists k0 ∈ N
+such that
+sup
+x∈R
+max
+i∈{0,...,8}[g(i)(x)/(1 + |x|k0)] < ∞ .
+Note that under A0 and A1, [14, Lemma A.1] implies that
+�
+R xk exp(−g(x))dx < ∞ for any
+k ∈ N. We also assume that the sequence of proximal MALA algorithms is initialized at stationarity.
+A2. For any d ∈ N∗, Xd
+0 has distribution πd.
+The assumptions above closely resemble those of [34] used to obtain the optimal scaling results
+for MALA. In particular, A1 ensures that we can approximate the log-acceptance ratio in (6) with
+a Taylor expansion, while A2 avoids technical complications due to the transient phase of the
+algorithm. We discuss how the latter assumption could be relaxed in Section 5.
+For technical reasons, and to allow direct comparisons with the results established in [34] for
+MALA, we will also consider the following regularity assumption
+A3. The function g′ is Lipschitz continuous.
+We denote by Ld
+t the linear interpolation of the first component of the discrete time Markov
+chain (Xd
+k)k≥0 obtained with the generic proximal MALA algorithm described above
+Ld
+t = (⌈d2αt⌉ − d2αt)Xd
+⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)Xd
+⌈d2αt⌉,1 ,
+(11)
+8
+
+where ⌊·⌋ and ⌈·⌉ denote the lower and upper integer part functions, respectively, and denote by
+Xd
+k,1 the first component of Xd
+k. The following result shows that in the limit d → ∞ the properties
+of proximal MALA depend on the relative speed at which σ2
+d = ℓ2/d2α and λd = c2/2d2β converge
+to 0. Recall that we set r = c2/ℓ2m > 0 and under A2, consider for any d ∈ N∗,
+ad(ℓ, r) = E
+� πd(Y d
+1 )qd(Y d
+1 , Xd
+0)
+πd(Xd
+0)qd(Xd
+0, Y d
+1 ) ∧ 1
+�
+.
+(12)
+Theorem 1. Assume A0, A1 and A2. For any d ∈ N∗, let σ2
+d = ℓ2/d2α and λd = c2/2d2β with
+α, β > 0. Then, the following statements hold.
+(a) If α = 1/4, β = 1/8 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ
+�
+−ℓ2K1(r)/2
+�
+, where Φ is the
+distribution function of a standard normal and
+K2
+1(r) = r2
+4 E
+��
+g′′(Xd
+0,1)g′(Xd
+0,1)
+�2�
+.
+If in addition, A3 holds.
+(b) If α = 1/6, β = 1/6 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ
+�
+−ℓ3K2(r)/2
+�
+, where Φ is the
+distribution function of a standard normal and
+K2
+2(r) =
+�r
+8 + r2
+4
+�
+E
+�
+g′′(Xd
+0,1)g′(Xd
+0,1)
+2�
++
+� 1
+16 + r
+8
+�
+E
+�
+g′′(Xd
+0,1)3�
++ 5
+48E
+�
+g′′′(Xd
+0,1)2�
+.
+(c) If α = 1/6, β > 1/6 and r > 0, we have limd→+∞ ad(ℓ, r) = 2Φ
+�
+−ℓ3K2(0)/2
+�
+, where Φ is the
+distribution function of a standard normal.
+In addition, in all these cases, as d → ∞ the process (Ld
+t )t≥0 converges weakly to the Langevin
+diffusion
+dLt = h(ℓ, r)1/2dBt − h(ℓ, r)
+2
+g′(x)dt ,
+(13)
+where (Bt)t≥0 denotes standard Brownian motion and h(ℓ, r) = ℓ2a(ℓ, r) is the speed of the diffusion,
+setting a(ℓ, r) = limd→∞ ad(ℓ, r). If α = 1/4, β = 1/8, for any r > 0, ℓ �→ h(ℓ, r) is maximized at
+the unique value of ℓ such that a(ℓ, r) = 0.452; while if α = 1/6, β = m/6 with m ≥ 1 and r > 0,
+ℓ �→ h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.574.
+Proof. The proof follows that of [34, Theorem 1, Theorem 2] and is postponed to Appendix A.
+The theorem above shows that the relative speed at which λd converges to 0 influences the
+scaling of the resulting proximal algorithm. In case (c), m > 1 and λd decays with d at a faster
+rate than σ2
+d. This causes the proximity map (4) to collapse onto the identity and therefore the
+proposal (10) is arbitrarily close to that of MALA. The resulting scaling limit also coincides with
+that of MALA established in [34, Theorem 1, Theorem 2].
+9
+
+If λd and σ2
+d decay at the same rate (case (b)), the amount of gradient information provided by
+the proximity map is controlled by r. Comparing our result for case (b) with [34, Theorem 1] we
+find that
+K2
+2(0) = 1
+16E
+�
+g′′(Xd
+0,1)3�
++ 5
+48E
+�
+g′′′(Xd
+0,1)2�
+= K2
+MALA;
+thus, we have
+K2
+2(r) = K2
+2(0) +
+�r
+8 + r2
+4
+�
+E
+�
+g′′(Xd
+0,1)2g′(Xd
+0,1)2�
++ r
+8E
+�
+g′′(Xd
+0,1)3�
+= K2
+MALA +
+�r
+8 + r2
+4
+�
+E
+�
+g′′(Xd
+0,1)2g′(Xd
+0,1)2�
++ r
+8E
+�
+g′′(Xd
+0,1)3�
+≥ K2
+MALA ,
+since the convexity of g implies that g′′ ≥ 0. In particular, K2
+2(r) is an increasing function of r
+achieving its minimum when r → 0 (i.e. MALA), see Figure 1(a).
+In case (a), m = 1/2 and λd decays more slowly than σ2
+d.
+As a consequence, the gradient
+information provided by the proximity map is smaller than in cases (b)–(c), and the resulting
+scaling differs from that of MALA. The value of K2
+1(r) is increasing in r and the speed of the
+corresponding diffusion also depends on r (see Figure 1(a) gray lines and Figure 1(b)).
+Example 1 (Gaussian target). Take g(x) = x2/2, proxg
+λ(x) = x/(1 + λ). In this case, g′ is Lipschitz
+continuous and we have K2
+1(r) = r2/4, K2
+2(r) =
+�
+1 + 4r + 4r2�
+/16 and K2
+2(0) = K2
+MALA = 1/16.
+The corresponding speeds are given in Figure 1(a). Optimizing for m = 1, r = 0 (MALA) and
+m = 1, r = 1 (P-MALA) we obtain
+hMALA(ℓ, r) = 1.5639,
+hP-MALA(ℓ, r) = 0.7519,
+achieved with ℓMALA = 1.6503 and ℓP-MALA = 1.1443, respectively. For Gaussian targets, MALA
+is geometrically ergodic [13], and therefore the optimal choice in terms of speed of convergence is
+MALA which is obtained for r = 0. The result for r = 1 and m = 1 are also given in [31, Theorem
+2.1].
+Example 2 (Target with light tails). Take g(x) = x4, which gives a normalized distribution with
+normalizing constant 2Γ(5/4). The proximity map is
+proxλ
+g(x) = 1
+2
+�
+3�
+9λ2x +
+√
+54λ4x2 + 3λ3
+32/3λ
+−
+1
+3�
+27λ2x + 3
+√
+54λ4x2 + 3λ3
+�
+.
+In this case g′ is not Lipschitz continuous and therefore we only consider (a), for which we have
+K2
+1(r) = 144r2Γ(11/4)/Γ(5/4). The corresponding speed is given in Figure 1(b).
+3.2
+Laplace target
+As discussed in the introduction, proximal MALA has been widely used to quantify uncertainty in
+imaging applications, in which target distributions involving the ℓ1 norm are particularly common
+[30, 14, 1, 46].
+Here, we consider πL
+d to be the product of d i.i.d. Laplace distributions as in (7),
+πL
+d (xd) =
+d
+�
+i=1
+πL(xd
+i ), for xd ∈ Rd, where πL(x) = 2−1 exp(−|x|) .
+(14)
+10
+
+r
+Speed
+(a) Gaussian target
+r
+Speed
+(b) Light tail target
+Figure 1: Value of K for i = 1, 2 and speed of the corresponding Langevin diffusion as a function
+of r for a Gaussian target and a light tail target. We denote by h1 the speed obtained in case (a),
+by h2 that obtained in (b). In case (c) both K3 and the speed h3 are constant w.r.t. r and coincide
+with that of MALA. For the Gaussian target we report the results for case (a)–(c) while for the
+light tail target we only report (a).
+For this particular choice of one-dimensional target distribution, the corresponding potential G is
+x �→ |x| and satisfies A0. Then, the proximity map is given by the soft thresholding operator [29,
+Section 6.1.3]
+proxλ
+G(x) = (x − sgn(x)λ)1{|x| > λ} ,
+(15)
+where sgn : R → {−1, 1} is the sign function, given by sgn(x) = −1 if x ≤ 0, and sgn(x) =
+1 otherwise. This operator is a continuous but not continuously differentiable map whose non-
+differentiability points are the extrema of the interval [−λ, λ] and are controlled by the value of the
+proximity parameter λ.
+Plugging (15) in (9), the proximal MALA algorithm applied to πL
+d proposes component-wise
+for i = 1, . . . , d
+Y d
+k+1,i = Xd
+k,i − σ2
+d
+2 sgn(Xd
+k,i)1{|Xd
+k,i| > λd} − σ2
+d
+2λd
+Xd
+k,i1{|Xd
+k,i| ≤ λd} + σdZd
+k+1,i .
+(16)
+For Xd
+k,i close to 0 (i.e. the point of non-differentiability) the proximal MALA proposal is a biased
+random walk around Xd
+k,i, while outside the region [−λd, λd] the proposal coincides with that of
+MALA. As λd → 0 the region in which the proximal MALA proposal coincide with that of MALA
+increases and when λd ≈ 0 the region [−λd, λd] in which the proposal corresponds to a biased
+random walk is negligible, as confirmed by the asymptotic acceptance rate in Theorem 2.
+We also consider the case λd = 0 for any d. Then, the proposal (16) becomes the proposal
+for the subgradient version of MALA: Y d
+k+1,i = Xd
+k,i − (σ2
+d/2) sgn(Xd
+k,i) + σdZd
+k+1,i, referred to as
+sG-MALA.
+The proof of the optimal scaling for the Laplace distribution follows the structure of that of
+[12] for Lp-mean differentiable distributions. We start by characterizing the asymptotic acceptance
+11
+
+ratio of a generic proximal MALA algorithm; contrary to Theorem 1 for differentiable targets, in
+the limit d → ∞ the properties of proximal MALA do not depend on the relative speed at which
+σ2
+d = ℓ2/d2α and λd = c2/2d2β converge to 0, as long as λd decays at least at the same rate as σ2
+d.
+In this regime, the region in which the proposal (16) corresponds to a biased random walk proposal
+is negligible, and therefore we obtain the same scaling obtained with λd = 0 and corresponding to
+sG-MALA.
+Theorem 2. Assume A2 and consider the sequence of target distributions {πL
+d }d∈N∗ given in (14).
+For any d ∈ N∗, let σ2
+d = ℓ2/d2α and λd = c2/2d2β with α = 1/3 and β = m/3 for m ≥ 1. Then, we
+have limd→∞ ad(ℓ, r) = aL(ℓ) = 2Φ(−ℓ3/2/(72π)1/4), where (ad(ℓ, r))d∈N∗ is defined in (12), with
+r = c2/ℓ2m, and Φ is the distribution function of a standard normal.
+Proof. The proof is postponed to Section 6.1.
+Note that the asymptotic average acceptance rate aL(ℓ) does not depend on r and as a result
+on c.
+Having identified the possible scaling for proximal MALA with Laplace target, we are now ready
+to show weak convergence to the appropriate Langevin diffusion. To this end, we adapt the proof
+strategy followed in [22] and [12].
+As for the differentiable case, consider the linear interpolation (Ld
+t )t≥0 of the first component
+of the Markov chain (Xd
+k)k≥0 given in (11). For any d ∈ N∗, denote by νd the law of the process
+(Ld
+t )t≥0 on the space of continuous functions from R+ to R, C(R+, R), endowed with the topology
+of uniform convergence over compact sets and its corresponding σ-field. We first show that the
+sequence (νd)d∈N∗, admits a weak limit point as d → ∞.
+Proposition 1. Assume A2 and consider the sequence of target distributions {πL
+d }d∈N∗ given
+in (14). For any d ∈ N∗, let σ2
+d = ℓ2/d2α and λd = c2/2d2β with α = 1/3 and β = m/3. The
+sequence (νd)d∈N∗ is tight in M1 (C(R+, R)), the set of probability measures acting on C(R+, R).
+Proof. See Section 6.2.
+By Prokhorov’s theorem, the tightness of (νd)d∈N∗ implies existence of a weak limit point ν. In
+our next result, we give a sufficient condition to show that any limit point of (νd)d∈N∗ coincides
+with the law of a solution of:
+dLt = [hL(ℓ)]1/2dBt − hL(ℓ)
+2
+sgn(Lt)dt .
+(17)
+To this end, we consider the martingale problem (see [42]) associated with (17), that we now
+present. Let us denote by C∞
+c (R, R) the subset of functions of C(R, R) which are infinitely many
+times differentiable and with compact support, and define the generator of (17) for V ∈ C∞
+c (R, R)
+by
+LV (x) = hL(ℓ)
+2
+[V ′′(x) − sgn(x)V ′(x)] .
+(18)
+Denote by (Wt)t≥0 the canonical process on C(R+, R), Wt : {ws}s≥0 �→ wt and the corresponding
+filtration by (Ft)t≥0. A probability measure ν is said to solve the martingale problem associated
+12
+
+with (17) with initial distribution πL, if the pushforward of ν by W0 is πL and if for all V ∈
+C∞
+c (R, R), the process
+�
+V (Wt) − V (W0) −
+� t
+0
+LV (Wu)du
+�
+t≥0
+is a martingale with respect to ν and the filtration (Ft)t≥0.
+The following proposition gives a
+sufficient condition to prove that ν is a solution of the martingale problem:
+Proposition 2. Suppose that for any V ∈ C∞
+c (R, R), m ∈ N, ρ : Rm → R bounded and continuous,
+and for any 0 ≤ t1 ≤ ... ≤ tm ≤ s ≤ t:
+lim
+d→+∞ Eνd
+��
+V (Wt) − V (Ws) −
+� t
+s
+LV (Wu)du
+�
+ρ(Wt1, ..., Wtm)
+�
+= 0 .
+Then any limit point of (νd)d∈N∗ on M1 (C(R+, R)) is a solution to the martingale problem associated
+with (17).
+Proof. See Section 6.3.
+Finally, we use this sufficient condition to establish that any limit point of (νd)d∈N∗ is a solution
+of the martingale problem for (17). Uniqueness in law of solutions of (17) allows to conclude that
+(Ld
+t )t≥0 converges weakly to the Langevin diffusion (17), which establishes our main result.
+Theorem 3. The sequence of processes {(Ld
+t )t≥0 : d ∈ N∗} converges in distribution towards
+(Lt)t≥0, solution of (17) as d → ∞, with hL(ℓ) = ℓ2aL(ℓ) and aL defined in Theorem 2.
+In
+addition, hL is maximized at the unique value of ℓ such that aL(ℓ) = 0.360.
+Proof. See Section 6.4.
+4
+Practical Implications and Numerical Simulations
+The optimal scaling results in Sections 3.1 and 3.2 provide some guidance on the choice of the
+parameters σ and λ of proximal MALA algorithms, suggesting that smaller values of λ provide
+better efficiency in terms of number of steps necessary to convergence (Theorem 1).
+However, a number of other factors must be taken into account. First, as shown in [26, 37, 36, 21]
+the convergence properties of Metropolis adjusted algorithms are influenced by the shape of the
+target distribution and, in particular, by its tail behavior. Secondly, when comparing proximal
+MALA algorithms with gradient-based methods (e.g. MALA) one must take into account the cost
+of obtaining the gradients, whether this comes from automatic differentiation algorithms or from
+evaluating a potentially complicated gradient function. On the other hand, proximity mappings can
+be quickly found or approximated solving convex optimization problems which have been widely
+studied in the convex optimization literature (e.g. [29, Chapter 6], [11] and [30, Section 3.2.3]).
+In terms of convergence properties, we are usually interested in the family of distributions for
+which the discrete time Markov chain produced by our algorithm is geometrically ergodic, together
+with the optimal scaling results briefly recalled in Section 2. Normally, the ergodicity results are
+given by considering the one-dimensional class of distributions E(β, γ) introduced in [36] and defined
+for γ > 0 and 0 < β < ∞ by
+E(β, γ) :
+�
+π : R → [0, +∞) : π(x) ∝ exp
+�
+−γ|x|β�
+, |x| > x0 for some x0 > 0
+�
+.
+13
+
+As observed by [24], there usually is a trade-off between ergodicity and optimal scaling results,
+algorithms providing better optimal scaling results tend to be geometrically ergodic for a smaller
+set of targets (e.g. MALA w.r.t. RWM).
+As suggested by Theorem 1, the scaling properties of proximal MALA on differentiable targets
+are close to those of MALA. This leads to a natural comparison between the two algorithms. First,
+we observe that A0 rules out targets for which G is not convex and therefore restricts the families
+E(β, γ) to β ≥ 1. To compare MALA with proximal MALA we therefore focus on distributions
+with β ≥ 1.
+It is shown in [36] that MALA is geometrically ergodic for targets in E(β, γ) with 1 ≤ β ≤ 2
+(with some caveat for β = 2). Theorem 1–(b) and (c) show that in this case proximal MALA
+has the same scaling properties of MALA but in case (b) the asymptotic speed of convergence
+decays as the constant r increases (Figure 1(a)), with the maximum achieved for r → 0, for which
+proximal MALA collapses onto MALA. Since MALA is geometrically ergodic, and achieves better
+(or equivalent) scaling properties than proximal MALA, it would be natural to prefer MALA to
+proximal MALA for this set of targets. However, if the gradient is costly to obtain, one might
+instead consider to use proximal MALA with a small λ, to retain scaling properties as close as
+possible to that of MALA but to reduce the computational cost of evaluating the gradient.
+In the case of differentiable targets with light-tails (i.e. β > 2), MALA is known not to be
+geometrically ergodic [36, Section 4.2] while the ergodicity properties of proximal MALA have only
+been partially studied in [30, Section 3.2.2] for the case λ = σ2/2 (P-MALA). As shown in [30,
+Section 2.1], given a distribution π ∈ E(β, γ) with β ≥ 1, the distribution πλ obtained using the
+potential (3) belongs to E(β′, γ′), where β′ = min(β, 2) and γ′ depending on λ. This suggests that
+proximal MALA is likely to be ergodic for appropriate choices of λ; a first result in this direction
+is given in [30, Corollary 3.2] for the P-MALA case λ = σ2/2. Theorem 1–(c) restricts the sets of
+available λs showing that for light-tail distributions (for which A3 does not hold) λ should decay
+at half the speed of σ2. Studying the ergodicity properties of proximal MALA in function of the
+parameter λ is, of course, an interesting problem that we leave for future work.
+For the Laplace distribution, Theorem 2 shows that the value of λ does not influence the
+asymptotic acceptance ratio of proximal MALA, as long as λ decays with d at least as fast as σ2.
+The scaling properties and the asymptotic speed h(ℓ) in Theorem 3 do not depend on λ and coincide
+with that of the sG-MALA (obtained for λ = 0). Hence, in terms of optimal scaling, there does not
+seem to be a difference between proximal MALA and sG-MALA for the Laplace distribution.
+4.1
+Numerical Experiments
+To illustrate the results established in Section 3.1 and 3.2 we consider here a small collection
+of simulation studies.
+The aim of these studies is to empirically confirm the optimal scalings
+identified in Theorem 1 and 2, investigate the dimension d at which the asymptotic acceptance
+ratio limd→∞ ad(ℓ, r) well approximates the empirical average acceptance ratio and, consequently,
+for which dimensions d we can expect the optimal asymptotic acceptances in Theorem 1 and 2 to
+guarantee maximal speed h(ℓ, r) (approximated by the expected squared jumping distance, see, e.g.
+[18]) for the corresponding diffusion. We summarize here our findings, a more detailed discussion
+can be found in Appendix B.
+For the differentiable case, we consider the Gaussian distribution in Example 1 and four algorith-
+mic settings which correspond to the three cases identified in Theorem 1 and MALA. The different
+values of r and m influence the dimension required to observe convergence to the theoretical limit
+14
+
+in Theorem 1: for r → 0 and m = 1 (MALA) and m = 1/2, r = 1 (corresponding to Theorem 1–(a))
+the theoretical limit is already achieved for d of order 102, while in the cases m = 3, r = 2 and
+m = r = 1 (corresponding to Theorem 1–(c) and (b), respectively) our simulation result match the
+theoretical limit only for d of order 105 or higher.
+The results for the Laplace case are similar, with the case m > 1 requiring a higher d to observe
+convergence to the theoretical limit.
+In general, we find that the optimal average acceptance ratios in Theorem 1 guarantee maximal
+speed h(ℓ, r) for d sufficiently large (for small d the optimal acceptance ratio often differs from the
+optimal asymptotic one, see, e.g. [40, Section 2.1]).
+5
+Discussion
+In this work we analyze the scaling properties of a wide class of proximal MALA algorithms intro-
+duced in [30, 14] for smooth targets and for the Laplace distribution. We show that the scaling
+properties of proximal MALA are influenced by the relative speed at which the proximal parameter
+λd and the proposal variance σd decay to 0 as d → ∞ and suggest practical ways to choose λd as a
+function of σd to guarantee good results.
+In the case of smooth targets, we provide a detailed comparison between proximal MALA and
+MALA, showing that proximal MALA scales no better than MALA (Theorem 1). In particular,
+Theorem 1–(a) shows that if λd is too large w.r.t. σd then the efficiency of proximal MALA is
+of order O(d−1/2) and therefore worse than the O(d−1/3) of MALA, suggesting that λd should
+be chosen to decay approximately as σd, if possible. If λd decays sufficiently fast, then MALA
+and proximal MALA have similar scaling properties and, in the case in which the proximity map
+is cheaper to compute that the gradient, one can build proximal MALA algorithms which are as
+efficient as MALA in terms of scaling but more computationally efficient.
+In the case of the Laplace distribution, we show that the scaling of proximal MALA is O(d−2/3)
+for any λd decaying sufficiently fast w.r.t. σd and, in the limit λd ≈ 0, we obtain a novel optimal
+scaling result for MALA on Laplace targets.
+As discussed in Section 4, our analysis provides some guidance on the choice of the parameters
+that need to be specified to implement proximal MALA, but this analysis should be complemented
+by an exploration of the ergodicity properties of proximal MALA to obtain a comprehensive descrip-
+tion of the algorithms. We conjecture that for sufficiently large values of λ, proximal MALA applied
+to light tail distributions will be exponentially ergodic; establishing exactly how large should λ be
+to guarantee fast convergence is an interesting question that we leave for future work. Obtaining
+these results would open the doors to adaptive tuning strategies for proximal MALA, which are
+likely to produce better results than those given by the strategies currently used.
+The set up under which we carried out our analysis closely resembles that of [34]; we anticipate
+that A2 could be relaxed following similar ideas as those in [10, 22] and that our analysis could be
+extended to d-dimensional targets πd possessing some dependence structure following the approach
+of [40, 4, 45]. Finally, the analysis carried out for the Laplace distribution could be extended to
+other piecewise smooth distributions provided that the moments necessary for the proof in Section 6
+can be computed.
+15
+
+6
+Proof of the Result for the Laplace distribution
+In this section we prove the results in Section 3.2 which give the scaling properties of proximal
+MALA (and sG-MALA) for the Laplace distribution. We collect technical results (e.g. moment
+computations, bounds, etc.) in Appendix D.
+We recall that σ2
+d = ℓ2/d2α and λd = c2/2d2β for some α, β > 0 and some constants c, ℓ
+independent on d.
+Thus, we can write λd as a function of σd, λd = σ2m
+d r/2, where we define
+r = c2/ℓ2m > 0 and m = β/α.
+In order to study the scaling limit of proximal MALA with Laplace target, consider the mapping
+bd : R2 → R given by
+bd : (x, z) �→ z − σd
+2 sgn(x)1
+�
+|x| > σ2m
+d r/2
+�
+−
+1
+σ2m−1
+d
+rx1
+�
+|x| ≤ σ2m
+d r/2
+�
+,
+(19)
+which allows us to write the proposal as Y d
+1,i = Xd
+0,i + σdbd(Xd
+0,i, Zd
+1,i) , for any i ∈ {1, . . . , d}.
+We consider also the function φd : R2 → R, given by
+φd : (x, z) �→ log π(x + σdbd(x, z))q(x + σdbd(x, z), x)
+π(x)q(x, x + σdbd(x, z))
+(20)
+= |x| − |x + σdbd(x, z)| + z2
+2
+−
+1
+2σ2
+d
+�σ2
+d
+2 sgn [x + σdbd(x, z)] 1
+�
+|x + σdbd(x, z)| > σ2m
+d r
+2
+�
+− σdbd(x, z)
++
+1
+σ2(m−1)
+d
+r
+[x + σdbd(x, z)] 1
+�
+|x + σdbd(x, z)| ≤ σ2m
+d r
+2
+��2
+.
+We introduce, for i ∈ {1, . . . , d}, φd,i = φd(Xd
+0,i, Zd
+1,i) for the sake of conciseness. This allows us to
+rewrite ad(ℓ, r), defined in (12), in the following way,
+ad(ℓ, r) = E
+�
+exp
+� d
+�
+i=1
+φd,i
+�
+∧ 1
+�
+.
+(21)
+Remark 1. Under A2, the families of random variables (bd(Xd
+0,i, Zd
+1,i))i∈{1,...,d} and (φd,i)i∈{1,...,d}
+are i.i.d.
+6.1
+Proof of Theorem 2
+The proof of Theorem 2 uses the first three moments of φd,1, whose computation is postponed to
+Appendix D.1, and is an application of Lindeberg’s central limit theorem.
+To identify the optimal scaling for the Laplace distribution, we look for those values of α such
+that �d
+i=1 E[φd,i] and Var(�d
+i=1 φd,i) converge to a finite value. Using Remark 1, we have that,
+d
+�
+i=1
+E [φd,i] = d E [φd,1]
+and
+Var
+� d
+�
+i=1
+φd,i
+�
+= d Var (φd,1) .
+(22)
+16
+
+Then, using the integrals in Appendix D.1, we find that the only value of α for which (22) converge to
+a finite value with the variance strictly positive is α = 1/3 as confirmed empirically in Appendix B.2.
+Having identified α = 1/3, we can then proceed applying Lindeberg’s CLT.
+Proof of Theorem 2. We start by showing that the acceptance ratio converges to a Gaussian dis-
+tribution. Define µd = E[φd,1] and Fd,i = σ((Xd
+0,j, Zd
+1,j), 1 ≤ j ≤ i), the natural filtration for
+(Xd
+0,i, Zd
+1,i)d∈N,1≤i≤d. The square-integrable martingale sequence
+�
+�Sd,i =
+i
+�
+j=1
+Wd,i, Fd,i
+�
+�
+d∈N∗,1≤i≤d
+where Wd,i = φd,i − µd, forms a triangular array, to which we can apply the corresponding CLT
+(e.g. [41, Theorem 4, page 543]). In particular, we have that,
+lim
+d→∞
+d
+�
+i=1
+E
+�
+W 2
+d,i | Fd,i−1
+�
+= lim
+d→∞ d Var (φd,1) =
+2ℓ3
+3
+√
+2π ,
+as shown in Proposition 17 in Appendix D.1. It remains to verify Lindeberg’s condition: for ε > 0,
+lim
+d→∞ dE
+�
+W 2
+d,11 {|Wd,1| > ε}
+�
+= 0 .
+In order to verify Lindeberg’s condition we verify the stronger Lyapunov condition: there exists
+ϵ > 0 such that
+lim
+d→∞ dE
+�
+W 2+ϵ
+d,1
+�
+= 0 .
+Pick ϵ = 1 and expand the cube using µd = E[φd,i],
+E
+�
+W 3
+d,1
+�
+= E
+�
+φ3
+d,i
+�
+− 3µdE
+�
+φ2
+d,i
+�
++ 2µ3
+d .
+(23)
+By Proposition 16 in Appendix D.1, we have limd→∞ dµ3
+d = 0, limd→∞ µd = 0, and, by Proposi-
+tion 17 in Appendix D.1,
+lim
+d→∞ dE
+�
+φ2
+d,i
+�
+=
+2ℓ3
+3
+√
+2π .
+Finally, for the remaining term in (23) we use Proposition 18 in Appendix D.1 to show that
+limd→∞ dE[φ3
+d,i] = 0. The above and the fact that, by Proposition 16 in Appendix D.1,
+lim
+d→∞ dµd = −
+ℓ3
+3
+√
+2π ,
+show, by Lindeberg’s CLT, that the acceptance ratio converges in law to a normal random variable
+�Z with mean −ℓ3/(3
+√
+2π) and variance 2ℓ3/(3
+√
+2π).
+To conclude the proof, we plug this convergence into (21), since x �→ ex ∧ 1 is a continuous and
+bounded mapping, we have that
+lim
+d→∞ exp
+� d
+�
+i=1
+φd,i
+�
+∧ 1
+d= e
+�
+Z ∧ 1
+and
+lim
+d→∞ ad(ℓ, r) = E
+�
+e
+�
+Z ∧ 1
+�
+,
+where the limit does not depend on r. Defining aL(ℓ) = limd→∞ ad(ℓ, r) and using [33, Proposition
+2.4], we have the result.
+17
+
+6.2
+Proof of Proposition 1
+We are interested in the law νd of the linear interpolant (Ld
+t )t≥0, defined in (11), of the first
+component of the chain (Xd
+k)k∈N. Let us recall the definition of the chain: assumption A2 gives
+the initial distribution πd, then, for any k ∈ N, the proposal Y d
+k+1 = (Y d
+k+1,i)1≤i≤d is defined in (16)
+with σ2
+d = ℓ2/d2α, λd = σ2m
+d r/2 with α = 1/3 and m ≥ 1. With the notations introduced in
+Section 6, we can rewrite (16), for any i ∈ {1, . . . , d},
+Y d
+k+1,i = Xd
+k,i + σdbd(Xd
+k,i, Zd
+k+1,i) ,
+(24)
+where bd is defined in (19) with r = c2/ℓ2m. From there, we apply the acceptance-rejection step
+described in (6), we additionally define the acceptance event Ad
+k+1 =
+�
+bd
+k+1 = 1
+�
+.
+We can now expand the expression of the linear interpolant Ld
+t , for t ≥ 0, using (6), (11) and
+the definition of Ad
+k+1,
+Ld
+t =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Xd
+⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 − σ2
+d
+2 sgn(Xd
+⌊d2αt⌋,1)
+�
+1Ad
+⌈d2αt⌉
+if |Xd
+⌊d2αt⌋,1| > σ2m
+d
+r
+2
+Xd
+⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 −
+1
+σ2(m−1)
+d
+rXd
+⌊d2αt⌋,1
+�
+1Ad
+⌈d2αt⌉
+otherwise
+,
+(25)
+or, equivalently,
+Ld
+t =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Xd
+⌈d2αt⌉,1 − (⌈d2αt⌉ − d2αt)
+�
+σdZd
+⌈d2αt⌉,1 − σ2
+d
+2 sgn(Xd
+⌊d2αt⌋,1)
+�
+1Ad
+⌈d2αt⌉
+if |Xd
+⌊d2αt⌋,1| > σ2m
+d
+r
+2
+Xd
+⌈d2αt⌉,1 − (⌈d2αt⌉ − d2αt)
+�
+σdZd
+⌈d2αt⌉,1 −
+1
+σ2(m−1)
+d
+rXd
+⌊d2αt⌋,1
+�
+1Ad
+⌈d2αt⌉
+otherwise
+.
+In order to prove Proposition 1, we consider Kolmogorov’s criterion for tightness (see [23, The-
+orem 23.7]): the sequence (νd)d≥1 is tight if
+E
+�
+(Ld
+t − Ld
+s)4�
+≤ γ(t)(t − s)2 ,
+for some non-decreasing positive function γ, all 0 ≤ s ≤ t and all d ∈ N∗ and the sequence (Ld
+0)d∈N∗
+is tight. The latter condition is straightforward to check, since by A2 the distribution of Ld
+0 = Xd
+0,1
+is πL for all d ∈ N∗.
+Proof of Proposition 1. Consider E
+�
+(Ld
+t − Ld
+s)4�
+, if ⌊d2αs⌋ = ⌊d2αt⌋, the inequality follows straight-
+forwardly recalling that the moments of normal distributions are bounded: in the case |Xd
+⌊d2αt⌋,1| =
+|Xd
+⌊d2αs⌋,1| > σ2m
+d r/2 it follows directly from the boundedness of the sgn function, while in the case
+|Xd
+⌊d2αt⌋,1| = |Xd
+⌊d2αs⌋,1| ≤ σ2m
+d r/2 we exploit the boundedness of Xd
+⌊d2αt⌋,1 itself.
+For all 0 ≤ s ≤ t such that ⌈d2αs⌉ ≤ ⌊d2αt⌋, we can distinguish three cases.
+18
+
+Case 1
+If |Xd
+⌊d2αt⌋,1| > σ2m
+d r/2 and |Xd
+⌊d2αs⌋,1| > σ2m
+d r/2, then
+Ld
+t − Ld
+s = Xd
+⌊d2αt⌋,1 − Xd
+⌈d2αs⌉,1
++ (d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 − σ2
+d
+2 sgn(Xd
+⌊d2αt⌋,1)
+�
+1Ad
+⌈d2αt⌉
++ (⌈d2αs⌉ − d2αs)
+�
+σdZd
+⌈d2αs⌉,1 − σ2
+d
+2 sgn(Xd
+⌊d2αs⌋,1)
+�
+1Ad
+⌈d2αs⌉ .
+Using H¨older’s inequality and the fact that 0 ≤ d2αt − ⌊d2αt⌋ ≤ 1 (and similarly for s) we have
+E
+�
+(Ld
+t − Ld
+s)4�
+≤ CE
+��
+Xd
+⌊d2αt⌋,1 − Xd
+⌈d2αs⌉,1
+�4�
++ C (d2αt − ⌊d2αt⌋)2
+d4α
+E
+��
+ℓZd
+⌈d2αt⌉,1
+�4
++
+ℓ4
+24d4α
+�
++ C (⌈d2αs⌉ − d2αs)2
+d4α
+E
+��
+ℓZd
+⌈d2αs⌉,1
+�4
++
+ℓ4
+24d4α
+�
+.
+Recalling that the moments of Zd are bounded and that d2αs ≤ ⌈d2αs⌉ ≤ ⌊d2αt⌋ ≤ d2αt, it follows
+E
+�
+(Ld
+t − Ld
+s)4�
+≤ C
+�
+(t − s)2 + E
+��
+Xd
+⌊d2αt⌋,1 − Xd
+⌈d2αs⌉,1
+�4��
+.
+(26)
+Case 2
+If |Xd
+⌊d2αt⌋,1| > σ2m
+d r/2 and |Xd
+⌊d2αs⌋,1| ≤ σ2m
+d r/2 or |Xd
+⌊d2αt⌋,1| ≤ σ2m
+d r/2 and |Xd
+⌊d2αs⌋,1| >
+σ2m
+d r/2. We only describe the argument for the first case, the second case follows from analogous
+steps. Take
+Ld
+t − Ld
+s = Xd
+⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 − σ2
+d
+2 sgn(Xd
+⌊d2αt⌋,1)
+�
+1Ad
+⌈d2αt⌉
+− Xd
+⌈d2αs⌉,1 − (⌈d2αs⌉ − d2αs)
+�
+σdZd
+⌈d2αs⌉,1 −
+1
+σ2(m−1)
+d
+r
+Xd
+⌊d2αs⌋,1
+�
+1Ad
+⌈d2αs⌉ .
+Proceeding as above, we find that
+E
+�
+(Ld
+t − Ld
+s)4�
+≤ C
+�
+(t − s)2 + E
+��
+Xd
+⌊d2αt⌋,1 − Xd
+⌈d2αs⌉,1
+�4�
++(⌈d2αs⌉ − d2αs)4E
+�
+�
+�
+1
+σ2(m−1)
+d
+r
+Xd
+⌊d2αs⌋
+�4�
+�
+�
+� ,
+and recalling that |Xd
+⌊d2αs⌋,1| ≤ σ2m
+d r/2 we have that |Xd
+⌊d2αs⌋,1|/(rσ2(m−1)
+d
+) ≤ σ2
+d/2. Using this
+and the same arguments as above, we have
+E
+�
+(Ld
+t − Ld
+s)4�
+≤ C
+�
+(t − s)2 + E
+��
+Xd
+⌊d2αt⌋,1 − Xd
+⌈d2αs⌉,1
+�4��
+.
+(27)
+19
+
+Case 3
+If |Xd
+⌊d2αt⌋,1| ≤ σ2m
+d r/2 and |Xd
+⌊d2αs⌋,1| ≤ σ2m
+d r/2, then
+Ld
+t − Ld
+s = Xd
+⌊d2αt⌋,1 + (d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 −
+1
+σ2(m−1)
+d
+r
+Xd
+⌊d2αt⌋,1
+�
+1Ad
+⌈d2αt⌉
+− Xd
+⌈d2αs⌉,1 + (⌈d2αs⌉ − d2αs)
+�
+σdZd
+⌈d2αs⌉,1 −
+1
+σ2(m−1)
+d
+r
+Xd
+⌊d2αs⌋,1
+�
+1Ad
+⌈d2αs⌉ .
+Using the boundedness of moments of Gaussian distributions and of Xd
+⌊d2αt⌋,1, Xd
+⌊d2αs⌋,1, we have
+E
+�
+(Ld
+t − Ld
+s)4�
+≤ C
+�
+(t − s)2 + E
+��
+Xd
+⌊d2αt⌋,1 − Xd
+⌈d2αs⌉,1
+�4��
+.
+(28)
+Putting (26), (27) and (28) together and using Lemma 1 below we obtain
+E
+��
+Ld
+t − Ld
+s
+�4�
+≤ C
+�
+(t − s)2 +
+4
+�
+p=2
+�
+⌊d2αt⌋ − ⌈d2αs⌉
+�p
+d2αp
+�
+≤ C(t − s)2 + C
+4
+�
+p=2
+d2αp (t − s)p
+d2αp
+≤ C
+�
+2 + t + t2�
+(t − s)2 ,
+which concludes the proof.
+We are now ready to state and prove Lemma 1:
+Lemma 1. There exists C > 0 such that for any k1, k2 ∈ N with 0 ≤ k1 < k2,
+E
+��
+Xd
+k2,1 − Xd
+k1,1
+�4�
+≤ C
+4
+�
+p=2
+(k2 − k1)p
+d2αp
+.
+Proof. Recalling the definition of the proposal in (24) and the notations of (19) we can write
+E
+��
+Xd
+k2,1 − Xd
+k1,1
+�4�
+= E
+�
+�
+�
+k2
+�
+k=k1+1
+σdbd
+�
+Xd
+k−1,1, Zd
+k,1
+�
+1Ad
+k
+�4�
+� .
+Then, we expand all acceptance or rejection terms between k1 and k2 and use H¨older’s inequality
+to obtain
+E
+��
+Xd
+k2,1 − Xd
+k1,1
+�4�
+≤ σ4
+dE
+�
+�
+�
+k2
+�
+k=k1+1
+bd
+�
+Xd
+k−1,1, Zd
+k,1
+�
+�4�
+�
++ σ4
+dE
+�
+�
+�
+k2
+�
+k=k1+1
+bd
+�
+Xd
+k−1,1, Zd
+k,1
+�
+1(Ad
+k)c
+�4�
+� ,
+20
+
+where bd is defined in (19). Using again H¨older’s inequality, for the first term we have
+E
+�
+�
+�
+k2
+�
+k=k1+1
+bd
+�
+Xd
+k−1,1, Zd
+k,1
+�
+�4�
+� ≤ C
+�
+�
+�E
+�
+�
+�
+k2
+�
+k=k1+1
+Zd
+k,1
+�4�
+�
++ σ4
+d
+24 E
+�
+�
+�
+k2
+�
+k=k1+1
+sgn
+�
+Xd
+k−1,1
+�
+1
+�
+|Xd
+k−1,1| > σ2m
+d r/2
+�
+�4�
+�
++σ4
+d
+24 E
+�
+�
+�
+k2
+�
+k=k1+1
+1
+σ2m−1
+d
+rXd
+k−1,11
+�
+|Xd
+k−1,1| ≤ σ2m
+d r/2
+�
+�4�
+�
+�
+�
+�
+≤ C
+�
+3(k2 − k1)2 + 2σ4
+d
+24 (k2 − k1)4
+�
+,
+(29)
+where the last line follows using the moments of Zd
+k,1 and the boundedness of Xd
+k−1,1 in the set
+{|Xd
+k−1,1| ≤ σ2m
+d r/2}.
+Using a Binomial expansion of the rejection term, we obtain
+E
+�
+�
+�
+k2
+�
+k=k1+1
+bd
+�
+Xd
+k−1,1, Zd
+k,1
+�
+1(Ad
+k)
+c
+�4�
+� =
+�
+E
+� 4
+�
+i=1
+bd
+�
+Xd
+mi−1,1, Zd
+mi,1
+�
+1(Admi)
+c
+�
+,
+(30)
+where the sum is over the quadruplets (mi)1≤i≤4 with mi ∈ {k1 + 1, . . . , k2}.
+We separate the terms in the sum according to their cardinality, let us denote, for j ∈ {1, . . . , 4},
+Ij =
+�
+(m1, . . . , m4) ∈ {k1 + 1, . . . , k2}4 : # {m1, . . . , m4} = j
+�
+;
+and define, for any (m1, . . . , m4) ∈ {k1 + 1, . . . , k2}4, �
+Xd
+0 = Xd
+0 and for any i ∈ {1, . . . , d},
+�
+Xd
+k+1,i = �
+Xd
+k,i + 1{m1−1,...,m4−1}c(k)1�Ad
+k+1σdbd
+�
+�
+Xd
+k,i, Zd
+k+1,i
+�
+,
+where
+�Ad
+k+1 =
+�
+Uk+1 ≤ exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+k,i, Zd
+k+1,i
+���
+,
+(31)
+and φd in (20). Denote by F the σ-algebra generated by the process �
+Xd and observe that on the
+event
+4�
+j=1
+�
+Ad
+mj
+�c
+, Xd is equal to �
+Xd.
+We consider now the terms in the sum (30).
+(i) If (m1, . . . , m4) ∈ I4, then the mis are all distinct and
+E
+�
+�
+4
+�
+j=1
+bd
+�
+Xd
+mj−1,1, Zd
+mj,1
+�
+1�
+Admj
+�c
+������
+F
+�
+� = E
+�
+�
+4
+�
+j=1
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�
+1�
+�Admj
+�c
+������
+F
+�
+� .
+21
+
+However, {bd( �
+Xd
+mj−1,1, Zd
+mj,1)1(�Ad
+mj )c}j=1,...,4 are independent conditionally on F. Thus,
+E
+�
+�
+4
+�
+j=1
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�
+1�
+�Admj
+�c
+������
+F
+�
+� =
+4
+�
+j=1
+E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�
+1�
+�Admj
+�c
+����F
+�
+=
+4
+�
+j=1
+E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�
+×
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+�
+,
+by integrating the uniform variables Umj in (31).
+Recalling the definition of bd in (19), we can bound the expectation above with
+������E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+� �
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+������
+(32)
+≤
+���E
+��σd
+2 sgn
+�
+�
+Xd
+mj−1,1
+�
+1
+�
+| �
+Xd
+mj−1,1| > σ2m
+d r/2
+�
+−
+1
+σ2m−1
+d
+r
+�
+Xd
+mj−1,11
+�
+| �
+Xd
+mj−1,1| ≤ σ2m
+d r/2
+��
+×
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+������
++
+�����E
+�
+Zd
+mj,1
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+������ .
+For the first one, we use the boundedness of the sgn function and of �
+Xd
+mj−1,1 in the set
+{| �
+Xd
+mj−1,1| ≤ σ2m
+d r/2} to obtain
+���E
+��σd
+2 sgn
+�
+�
+Xd
+mj−1,1
+�
+1
+�
+| �
+Xd
+mj−1,1| > σ2m
+d r/2
+�
+−
+1
+σ2m−1
+d
+r
+�
+Xd
+mj−1,11
+�
+| �
+Xd
+mj−1,1| ≤ σ2m
+d r/2
+��
+×
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+������
+≤ σd
+2 E
+������
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����
+�����F
+�
+≤ σd
+2 .
+(33)
+We can write the second term as
+E
+�
+Zd
+mj,1
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+�
+= E
+�
+G
+�
+�
+Xd
+mj−1,1,
+d
+�
+i=2
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+�������F
+�
+,
+22
+
+where we define G(a, b) = E
+�
+Z (1 − exp (φd (a, Z) + b))+
+�
+with Z a standard Gaussian. Be-
+cause the function x �→ (1 − exp(x))+ is 1-Lipschitz, we have, using Cauchy-Schwarz and
+Lemma 3 in Appendix D.2,
+��E
+�
+Z (1 − exp (φd (a, Z) + b))+
+�
+− E
+�
+Z (1 − exp (b))+
+��� ≤ E [|Z| |φd (a, Z)|]
+≤ E
+�
+Z2�1/2 E
+�
+φd (a, Z)2�1/2
+≤ E
+�
+φd (a, Z)2�1/2
+≤ Cd−α .
+However, E
+�
+Z (1 − exp (b))+
+�
+= E [Z] (1 − exp (b))+ = 0, and therefore
+�����E
+�
+G
+�
+�
+Xd
+mj−1,1,
+d
+�
+i=2
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+�������F
+������ ≤ Cd−α .
+(34)
+Combining equations (32), (33) and (34) and recalling that σd = ℓd−α, we have
+�����E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+� �
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+������ ≤ Cd−α ,
+(35)
+from which follows that
+�
+(m1,...,m4)∈I4
+�����E
+� 4
+�
+i=1
+bd
+�
+Xd
+mi−1,1, Zd
+mi,1
+�
+1(Admi)
+c
+������ ≤
+�
+(m1,...,m4)∈I4
+E
+�
+�
+4
+�
+j=1
+C
+dα
+�
+�
+≤
+�k2 − k1
+4
+� C
+d4α ≤ C (k2 − k1)4
+d4α
+,
+(36)
+using that |I4| =
+�k2−k1
+4
+�
+.
+(ii) If (m1, .., m4) ∈ I3, only three of the mis take distinct values; proceeding as in case (i), we
+have
+������
+E
+�
+�
+3
+�
+j=1
+bd
+�
+Xd
+mj−1,1, Zd
+mj,1
+�1+δ1,j
+1�
+Admj
+�c
+������
+F
+�
+�
+������
+=
+3
+�
+j=1
+�����E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�1+δ1,j
+�
+1 − exp
+� d
+�
+i=1
+φd
+�
+�
+Xd
+mj−1,i, Zd
+mj,i
+���
++
+�����F
+������ ,
+where δ1,j denotes a Dirac’s delta. For the terms j ̸= 1, we use (35), while for the term j = 1
+23
+
+we bound the indicator function by 1 to obtain
+������
+E
+�
+�
+3
+�
+j=1
+bd
+�
+Xd
+mj−1,1, Zd
+mj,1
+�1+δ1,j
+1�
+Admj
+�c
+������
+F
+�
+�
+������
+≤
+����E
+�
+bd
+�
+�
+Xd
+m1−1,1, Zd
+m1,1
+�2����F
+�����
+3
+�
+j=2
+C
+dα
+≤
+�
+3 + 2σ2
+d
+22d2α
+� C2
+d2α ≤ C 1
+d2α ,
+where the second-to-last inequality follows using the same approach taken for (29) and recall-
+ing that σd = ℓd−α. Hence,
+�
+(m1,...,m4)∈I3
+�����E
+� 4
+�
+i=1
+bd
+�
+Xd
+mi−1,1, Zd
+mi,1
+�
+1(Admi)
+c
+������
+(37)
+≤ C
+�k2 − k1
+3
+� 1
+d2α ≤ C (k2 − k1)3
+d2α
+.
+(iii) If (m1, .., m4) ∈ I2, we have two different cases: the mis take the two values twice or three
+mis have the same value. For the first one, we have, bounding the indicator function with 1,
+E
+�
+�E
+�
+�
+2
+�
+j=1
+bd
+�
+Xd
+mj−1,1, Zd
+mj,1
+�2
+1�
+Admj
+�c
+������
+F
+�
+�
+�
+�
+≤ E
+�
+�
+2
+�
+j=1
+E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�2����F
+��
+� .
+Since, conditionally on F, the random variables inside the expectation are Gaussians with
+bounded mean and variance 1, we have, using the same approach taken for (29),
+E
+�
+�
+2
+�
+j=1
+E
+�
+bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+�2����F
+��
+� ≤
+�
+1 + 2σ2
+d
+22
+�2
+≤ C .
+The second case follows similarly
+������
+E
+�
+�E
+�
+�
+2
+�
+j=1
+bd
+�
+Xd
+mj−1,1, Zd
+mj,1
+�1+2δ1,j
+1�
+Admj
+�c
+������
+F
+�
+�
+�
+�
+������
+≤ E
+�
+�E
+�
+�
+2
+�
+j=1
+���bd
+�
+�
+Xd
+mj−1,1, Zd
+mj,1
+����
+1+2δ1,j
+������
+F
+�
+�
+�
+� ≤ C ,
+24
+
+where δ1,j denotes a Dirac’s delta. Therefore,
+�
+(m1,...,m4)∈I2
+�����E
+� 4
+�
+i=1
+�
+bd(Xd
+mi−1,1, Zd
+mi,1
+�
+1(Admi)
+c
+������
+(38)
+≤ C
+��4
+2
+�
++
+�4
+3
+�� �k2 − k1
+2
+�
+≤ C(k2 − k1)2 .
+(iv) If (m1, .., m4) ∈ I1 (i.e. all mis take the same value), we bound the indicator function by 1
+and, using the same approach taken for (29), we find
+E
+�
+bd
+�
+Xd
+m1−1,1, Zd
+m1,1
+�4 1(Adm1)
+c
+�
+≤ C
+�
+3 + 2σ4
+d
+24
+�
+≤ C ,
+since σd = ℓd−α and d ∈ N. Hence,
+�
+(m1,...,m4)∈I1
+�����E
+� 4
+�
+i=1
+bd
+�
+Xd
+m1−1,1, Zd
+m1,1
+�
+1(Ad
+mi)
+c
+������ ≤ C
+�k2 − k1
+1
+�
+= C(k2 − k1) .
+(39)
+The result follows combining (36), (37), (38) and (39) in (30).
+6.3
+Proof of Proposition 2
+We start by proving the following lemma.
+Lemma 2. Let ν be a limit point of the sequence of laws (νd)d≥1 of {(Ld
+t )t≥0 | d ∈ N}. Then for
+any t ≥ 0, the pushforward measure of ν by Wt is πL(dx) = exp(−|x|)dx/2.
+Proof. Using (25), we have
+E
+����Ld
+t − Xd
+⌊d2αt⌋,1
+���
+�
+≤ E
+�����(d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 − σ2
+d
+2 sgn(Xd
+⌊d2αt⌋,1)
+�
+1|Xd
+⌊d2αt⌋,1|>σ2m
+d
+r/21Ad
+⌈d2αt⌉
+����
+�
++ E
+������(d2αt − ⌊d2αt⌋)
+�
+σdZd
+⌈d2αt⌉,1 −
+1
+σ2(m−1)
+d
+r
+Xd
+⌊d2αt⌋,1
+�
+1|Xd
+⌊d2αt⌋,1|≤σ2m
+d
+r/21Ad
+⌈d2αt⌉
+�����
+�
+≤ (d2αt − ⌊d2αt⌋)
+�
+σdE
+����Zd
+⌈d2αt⌉,1)
+���
+�
++ σ2
+d
+2 E
+����sgn(Xd
+⌊d2αt⌋,1)
+���
+�
++
+1
+σ2(m−1)
+d
+r
+E
+����Xd
+⌊d2αt⌋,1
+��� 1|Xd
+⌊d2αt⌋,1|≤σ2m
+d
+r/2
+��
+≤ (d2αt − ⌊d2αt⌋)
+�
+ℓ
+dα E
+�
+(Zd
+⌈d2αt⌉,1)2�1/2
++
+ℓ2
+2d2α +
+1
+σ2(m−1)
+d
+r
+E
+�σ2m
+d r
+2
+��
+≤ (d2αt − ⌊d2αt⌋)
+� ℓ
+dα +
+ℓ2
+2d2α +
+ℓ2
+2d2α
+�
+≤ C
+dα ,
+25
+
+where we used Cauchy-Schwarz inequality and the fact that the moments of Zd
+⌈d2αt⌉,1 are bounded.
+The above guarantees that,
+lim
+d→∞ E
+����Ld
+t − Xd
+⌊d2αt⌋,1
+���
+�
+= 0 .
+As (νd)d≥1 converges weakly towards ν, for any Lipschitz bounded function ψ : R → R,
+lim
+d→∞ E
+�
+ψ
+�
+Xd
+⌊d2αt⌋,1
+��
+= lim
+d→∞ E
+�
+ψ
+�
+Ld
+t
+��
+= Eν [ψ(Wt)] .
+The result follows since Xd
+⌊d2αt⌋,1 is distributed according to πL(dx) = exp(−|x|)dx/2 for any t ≥ 0
+and d ∈ N.
+We are now ready to prove Proposition 2:
+Proof of Proposition 2. Let ν be a limit point of (νd)d≥1.
+We start by showing that if for any
+V ∈ C∞
+c (R, R), m ∈ N, any bounded and continuous mapping ρ : Rm → R and any 0 ≤ t1 ≤ · · · ≤
+tm ≤ s ≤ t, ν satisfies
+Eν
+��
+V (Wt) − V (Ws) −
+� t
+s
+LV (Wu)du
+�
+ρ(Wt1, . . . , Wtm)
+�
+= 0 ,
+(40)
+then ν is a solution to the martingale problem associated with L.
+Let Fs denote the σ-algebra generated by
+{ρ(Wt1, . . . , Wtm) : m ∈ N, ρ : Rm → R bounded and continuous, and 0 ≤ t1 ≤ · · · ≤ tm ≤ s} .
+Then,
+Eν
+�
+V (Wt) − V (Ws) −
+� t
+s
+LV (Wu)du
+����Fs
+�
+= 0 ,
+showing that the process
+�
+V (Wt) − V (W0) −
+� t
+0
+LV (Wu)du
+�
+t≥0
+is a martingale w.r.t. ν and the filtration (Ft)t≥0.
+To prove (40), it is enough to show that for any V ∈ C∞
+c (R, R), m ∈ N and any bounded and
+continuous mapping ρ : Rm → R and any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t, the mapping
+Ψs,t : w �−→
+�
+V (wt) − V (ws) −
+� t
+s
+LV (wu)du
+�
+ρ (wt1, . . . , wtm) ,
+is continuous on a ν-almost sure subset of C(R+, R). Let
+W = {w ∈ C(R+, R) : wu ̸= 0 for almost any u ∈ [s, t]} .
+Since w ∈ Wc if and only if
+� t
+s 1{0}(wu)du > 0, using Lemma 2 and the Fubini–Tonelli’s theorem,
+Eν
+�� t
+s
+1{0}(Wu)du
+�
+=
+� t
+s
+Eν �
+1{0}(Wu)
+�
+du =
+� t
+s
+πL({0})du = 0 ,
+26
+
+and we have that ν(Wc) = 0.
+Since w �→ wu is continuous for any u ≥ 0, so are w �→ V (wu) and w �→ ρ(wt1, . . . , wtm).
+Thus, it is enough to prove that the mapping w �→
+� t
+s LV (wu)du is continuous.
+Let (wn)n≥0
+be a sequence in C(R+, R) that converges to w ∈ W in the uniform topology on compact sets.
+Let u be such that wu ̸= 0, therefore, since the sgn function is continuous in a neighbourhood
+of wu, limn→∞ LV (wn
+u) = LV (wu), thus limn→∞ LV (wn
+u) = LV (wu) for almost any u ∈ [s, t].
+Finally, using the boundedness of the sequence (LV (wn
+u))n≥0 and Lebesgue’s dominated convergence
+theorem,
+lim
+n→∞
+� t
+s
+LV (wn
+u)du =
+� t
+s
+LV (wu)du ,
+which proves that the mappings Ψs,t are continuous on W.
+6.4
+Proof of Theorem 3
+Let us introduce, for any n ∈ N, Fd
+n,1 = σ({Xd
+k,1, 0 ≤ k ≤ n}), the σ-algebra generated by the first
+components of {Xd
+k | 0 ≤ k ≤ n}. We also introduce for any V ∈ C∞
+c (R, R)
+M d
+n(V ) = ℓ
+dα
+n−1
+�
+k=0
+V ′(Xd
+k,1)
+×
+�
+bd
+�
+Xd
+k,1, Zd
+k+1,1
+�
+1Ad
+k+1 − E
+�
+bd
+�
+Xd
+k,1, Zd
+k+1,1
+�
+1Ad
+k+1
+���Fd
+k,1
+��
+(41)
++
+ℓ2
+2d2α
+n−1
+�
+k=0
+V ′′(Xd
+k,1)
+×
+�
+bd
+�
+Xd
+k,1, Zd
+k+1,1
+�2 1Ad
+k+1 − E
+�
+bd
+�
+Xd
+k,1, Zd
+k+1,1
+�2 1Ad
+k+1
+���Fd
+k,1
+��
+.
+where bd is defined in (19).
+The proof of Theorem 3 follows using the sufficient condition in Proposition 2, the tightness of
+the sequence (νd)d≥1 established in Proposition 1 and Proposition 3 below.
+Proof. Using Proposition 1, Proposition 2 and Proposition 3 below, it is enough to show that for
+any V ∈ C∞
+c (R, R), m ≥ 1, any 0 ≤ t1 ≤ · · · ≤ tm ≤ s ≤ t and any bounded and continuous
+mapping ρ : Rm → R,
+lim
+d→∞ E
+��
+M d
+⌈d2αt⌉(V ) − M d
+⌈d2αs⌉(V )
+�
+ρ(Ld
+t1, ..., Ld
+tm)
+�
+= 0 ,
+where, for any n ≥ 1, M d
+n(V ) is given by (41). However, this is straightforwardly obtained by taking
+successively the conditional expectations with respect to Fd
+k,1 for k = ⌈d2αt⌉, . . . , ⌈d2αs⌉.
+Proposition 3. For any 0 ≤ s ≤ t, V ∈ Cc(R, R) we have
+lim
+d→∞ E
+�����V
+�
+Ld
+t
+�
+− V
+�
+Ld
+s
+�
+−
+� t
+s
+LV
+�
+Ld
+u
+�
+du −
+�
+M d
+⌈d2αt⌉ (V ) − M d
+⌈d2αs⌉ (V )
+�����
+�
+= 0 ,
+(42)
+where (Ld
+t )t≥0 is defined in (11).
+27
+
+Proof. The process (Ld
+t )t≥0 is piecewise linear, thus it has finite variation. For any τ ≥ 0, we define
+dLd
+τ = d2ασdbd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1)
+�
+1Ad
+⌈d2ατ⌉dτ .
+Thus, recalling that σd = ℓd−α and using the fundamental theorem of integral calculus for piecewise
+C1 maps
+V
+�
+Ld
+t
+�
+− V
+�
+Ld
+s
+�
+= ℓdα
+� t
+s
+V ′ �
+Ld
+τ
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉dτ ,
+(43)
+where bd is defined in (19). A Taylor expansion of V ′ with Lagrange remainder about Xd
+⌊d2ατ⌋,1
+gives
+V ′ �
+Ld
+τ
+�
+= V ′ �
+Xd
+⌊d2ατ⌋,1
+�
++ ℓ
+dα
+�
+d2ατ − ⌊d2ατ⌋
+�
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉
++
+ℓ2
+2d2α
+�
+d2ατ − ⌊d2ατ⌋
+�2 V (3) (χτ) bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉ ,
+where for any point τ ∈ [s, t], there exists χτ ∈ [Xd
+⌊d2ατ⌋,1, Y d
+τ,1]. Substituting the above into (43)
+we obtain
+V
+�
+Ld
+t
+�
+− V
+�
+Ld
+s
+�
+= ℓdα
+� t
+s
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉dτ
++ ℓ2
+� t
+s
+�
+d2ατ − ⌊d2ατ⌋
+�
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉dτ
+(44)
++ ℓ3
+2dα
+� t
+s
+�
+d2ατ − ⌊d2ατ⌋
+�2 V (3) (χτ) bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�3
+1Ad
+⌈d2ατ⌉dτ .
+Since V (3) is bounded, using Fubini-Tonelli’s theorem and recalling the definition of bd in (19), we
+have that
+ℓ3
+2dα E
+�����
+� t
+s
+�
+d2ατ − ⌊d2ατ⌋
+�2 V (3) (χτ) bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�3
+1Ad
+⌈d2ατ⌉dτ
+����
+�
+≤ C ℓ3
+2dα
+� t
+s
+E
+�����Zd
+⌈d2ατ⌉,1
+��� +
+ℓ
+2dα
+�3�
+dτ −→
+d→∞ 0 ,
+since the moments of Zd
+⌈d2ατ⌉,1 are bounded.
+For the second term in (44), we observe that most of the integrand is piecewise constant since
+the process Xd
+⌊d2ατ⌋,1 evolves in discrete time. Then, for any integer d2αs ≤ k ≤ d2αt − 1,
+� (k+1)/d2α
+k/d2α
+�
+d2ατ − ⌊d2ατ⌋
+�
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉dτ
+=
+1
+2d2α V ′′ �
+Xd
+k,1
+�
+bd
+�
+Xd
+k,1, Zd
+k+1,1
+�2 1Ad
+k+1
+= 1
+2
+� (k+1)/d2α
+k/d2α
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉dτ .
+28
+
+Thus, we can write
+I =
+� t
+s
+�
+d2ατ − ⌊d2ατ⌋
+�
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉dτ
+= I1 + I2 ,
+where we define
+I2 = 1
+2
+� t
+s
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉dτ ,
+and
+I1 =
+�� ⌈d2αs⌉/d2α
+s
++
+� t
+⌊d2αt⌋/d2α
+� �
+d2ατ − ⌊d2ατ⌋ − 1
+2
+�
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+× bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉dτ .
+In addition, we have
+I1 =
+1
+2d2α
+�
+d2αs ��� ⌊d2αs⌋
+� �
+⌈d2αs⌉ − d2αs
+�
+V ′′ �
+Xd
+⌊d2αs⌋,1
+�
+bd
+�
+Xd
+⌊d2αs⌋,1, Zd
+⌈d2αs⌉,1
+�2
+1Ad
+⌈d2αs⌉
++
+1
+2d2α
+�
+d2αt − ⌊d2αt⌋
+� �
+⌈d2αt⌉ − d2αt
+�
+V ′′ �
+Xd
+⌊d2αt⌋,1
+�
+bd
+�
+Xd
+⌊d2αt⌋,1, Zd
+⌈d2αt⌉,1
+�2
+1Ad
+⌈d2αt⌉ ,
+and, since V ′′ and the moments of Zd
+⌈d2αt⌉,1 are bounded, limd→∞ E [|I1|] = 0. Thus,
+lim
+d→∞ E
+���V
+�
+Ld
+t
+�
+− V
+�
+Ld
+s
+�
+− Is,t
+���
+= 0 ,
+where
+Is,t =
+� t
+s
+�
+ℓdαV ′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+(45)
++ ℓ2
+2 V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉
+�
+dτ .
+Next, we use (18) and write
+� t
+s
+LV
+�
+Ld
+τ
+�
+dτ =
+� t
+s
+hL(ℓ)
+2
+�
+V ′′ �
+Xd
+⌊d2ατ⌋1
+�
+− sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+V ′ �
+Xd
+⌊d2ατ⌋,1
+��
+dτ − T d
+3 ,
+(46)
+where we define
+T d
+3 =
+� t
+s
+�
+LV
+�
+Xd
+⌊d2ατ⌋,1
+�
+− LV
+�
+Ld
+τ
+��
+dτ .
+Finally, we write the difference M d
+⌈d2αt⌉(V ) − M d
+⌈d2αs⌉(V ) as the integral of a piecewise constant
+29
+
+function
+M d
+⌈d2αt⌉(V ) − M d
+⌈d2αs⌉(V ) = Is,t
+(47)
+−
+� t
+s
+�
+ℓdαV ′ �
+Xd
+⌊d2ατ⌋,1
+�
+E
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
++ℓ2
+2 V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+E
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉
+����Fd
+⌊d2ατ⌋,1
+��
+dτ
+− T d
+4 − T d
+5 ,
+where T d
+4 and T d
+5 account for the difference between the sum in (41) and the integral, and are
+defined as
+T d
+4 = − ℓ
+dα
+�
+⌈d2αt⌉ − d2αt
+�
+V ′ �
+Xd
+⌊d2αt⌋,1
+� �
+bd
+�
+Xd
+⌊d2αt⌋,1, Zd
+⌈d2αt⌉,1
+�
+1Ad
+⌈d2αt⌉
+−E
+�
+bd
+�
+Xd
+⌊d2αt⌋,1, Zd
+⌈d2αt⌉,1
+�
+1Ad
+⌈d2αt⌉
+���Fd
+⌊d2αt⌋,1
+��
+−
+ℓ2
+2d2α
+�
+⌈d2αt⌉ − d2αt
+�
+V ′′ �
+Xd
+⌊d2αt⌋,1
+� �
+bd
+�
+Xd
+⌊d2αt⌋,1, Zd
+⌈d2αt⌉,1
+�2
+1Ad
+⌈d2αt⌉
+−E
+�
+bd
+�
+Xd
+⌊d2αt⌋,1, Zd
+⌈d2αt⌉,1
+�2
+1Ad
+⌈d2αt⌉
+����Fd
+⌊d2αt⌋,1
+��
+,
+and T d
+5 = −T d
+4 with t substituted by s.
+Putting (45), (46) and (47) together we obtain
+Is,t −
+� t
+s
+LV
+�
+Ld
+τ
+�
+dτ −
+�
+M d
+⌈d2αt⌉(V ) − M d
+⌈d2αs⌉(V )
+�
+= T d
+1 + T d
+2 + T d
+3 + T d
+4 + T d
+5 ,
+where T d
+1 takes into account all the terms involving V ′(Xd
+⌊d2ατ⌋,1), and T d
+2 the terms involving
+V ′′(Xd
+⌊d2ατ⌋,1):
+T d
+1 =
+� t
+s
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+×
+�
+ℓdαE
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
++ hL(ℓ)
+2
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+��
+dτ ,
+T d
+2 =
+� t
+s
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+×
+�ℓ2
+2 E
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉
+����Fd
+⌊d2ατ⌋,1
+�
+− hL(ℓ)
+2
+�
+dτ .
+To obtain (42) it is then sufficient to prove that for any 1 ≤ i ≤ 5, limd→∞ E
+���T d
+i
+���
+= 0.
+Since V ′, V ′′ are bounded and bd is bounded in expectation because the moments of Zd
+⌈d2ατ⌉,1
+are bounded, it is easy to show that limd→∞ E
+���T d
+i
+���
+= 0 for i = 4, 5. For T d
+3 , we write T d
+3 =
+30
+
+hL(ℓ)(T d
+3,1 − T d
+3,2)/2, where
+T d
+3,1 =
+� t
+s
+�
+V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+− V ′′ �
+Ld
+τ
+��
+dτ ,
+T d
+3,2 =
+� t
+s
+�
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+− sgn
+�
+Ld
+τ
+�
+V ′ �
+Ld
+τ
+��
+dτ .
+Using Fubini-Tonelli’s theorem, the convergence of Xd
+⌊d2ατ⌋,1 to Ld
+τ in Lemma 2 and Lebesgue’s
+dominated convergence theorem we obtain
+E
+���T d
+3,1
+���
+≤
+� t
+s
+E
+����V ′′ �
+Xd
+⌊d2ατ⌋,1
+�
+− V ′′ �
+Ld
+τ
+����
+�
+dτ −→
+d→∞ 0 .
+We can further decompose T d
+3,2 as
+T d
+3,2 =
+� t
+s
+�
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+− sgn
+�
+Ld
+τ
+��
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+dτ
++
+� t
+s
+sgn
+�
+Ld
+τ
+� �
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+− V ′ �
+Ld
+τ
+��
+dτ .
+Proceeding as for T d
+3,1, it is easy to show that the second integral converges to 0 as d → ∞. We
+then bound the first integral by
+E
+�����
+� t
+s
+�
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+− sgn
+�
+Ld
+τ
+��
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+dτ
+����
+�
+≤ C
+� t
+s
+E
+����sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+− sgn
+�
+Ld
+τ
+����
+�
+dτ .
+However, since {sgn(Xd
+⌊d2ατ⌋,1) ̸= sgn(Ld
+τ)} ⊂ {sgn(Xd
+⌊d2ατ⌋,1) ̸= sgn(Xd
+⌈d2ατ⌉,1)}, using Lemma 4 in
+Appendix D.3 we have that
+E
+����sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+− sgn
+�
+Ld
+τ
+����
+�
+= 2E
+�
+1�
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+̸=sgn(Ldτ )
+�
+�
+= 2E
+�
+1�
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+̸=sgn
+�
+Xd
+⌈d2ατ⌉,1
+��
+�
+−→
+d→∞ 0 .
+The above and the dominated converge theorem show that
+E
+�����
+� t
+s
+�
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+− sgn
+�
+Ld
+τ
+��
+V ′ �
+Xd
+⌊d2ατ⌋,1
+�
+dτ
+����
+�
+−→
+d→∞ 0 .
+Consider then T d
+1 , recalling that the derivatives of V are bounded, we have
+E
+���T d
+1
+���
+≤
+� t
+s
+CE
+����ℓdαE
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
++ hL(ℓ)
+2
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�����
+�
+dτ
+≤
+� t
+s
+C
+�
+E
+����D(1)
+1,τ
+���
+�
++ E
+����D(1)
+2,τ
+���
+��
+dτ ,
+31
+
+where we define
+D(1)
+1,τ = ℓdαE
+�
+Zd
+⌈d2ατ⌉,11Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
+,
+D(1)
+2,τ = hL(ℓ)
+2
+sgn
+�
+Xd
+⌊d2ατ⌋,1
+�
+− ℓdα
+�σd
+2 sgn(Xd
+⌊d2ατ⌋,1)1|Xd
+⌊d2ατ⌋,1|>σ2m
+d
+r/2 +
+1
+σ2m−1
+d
+rXd
+⌊d2ατ⌋,11|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2
+�
+× E
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
+.
+Let us start with D(1)
+1,τ:
+D(1)
+1,τ = ℓdαE
+�
+Zd
+⌈d2ατ⌉,1
+�
+1 ∧ exp
+� d
+�
+i=1
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��������Fd
+⌊d2ατ⌋,1
+�
+,
+where φd is given in (20). Then, by independence of the components of Zd
+⌈d2ατ⌉, we have
+E
+�
+Zd
+⌈d2ατ⌉,1
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��������Fd
+⌊d2ατ⌋,1
+�
+= E
+�
+Zd
+⌈d2ατ⌉,1
+�
+E
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������Fd
+⌊d2ατ⌋,1
+�
+= 0 .
+This allows us to write
+E
+�
+|D(1)
+1,τ|
+�
+≤ ℓdαE
+�
+|Zd
+⌈d2ατ⌉,1|
+�����1 ∧ exp
+� d
+�
+i=1
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��
+− 1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������
+�
+.
+However, x �→ 1 ∧ exp(x) is a 1-Lipschitz function, thus
+E
+�
+|D(1)
+1,τ|
+�
+≤ ℓdαE
+�
+|Zd
+⌈d2ατ⌉,1|
+���φd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+����
+�
+,
+and D(1)
+1,τ → 0 as d → ∞ by Lemma 5 in Appendix D.3.
+For D(1)
+2,τ, we observe that
+− σd
+2 1|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2 ≤
+1
+σ2m−1rXd
+⌊d2ατ⌋,11|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2 ≤ σd
+2 1|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2 .
+(48)
+Distinguishing between Xd
+⌊d2ατ⌋,1 < 0 and Xd
+⌊d2ατ⌋,1 ≥ 0, it follows that
+|D(1)
+2,τ| ≤
+���sgn
+�
+Xd
+⌊d2ατ⌋,1
+����
+×
+����
+hL(ℓ)
+2
+− ℓdα �σd
+2 1|Xd
+⌊d2ατ⌋,1|>σ2m
+d
+r/2 + σd
+2 1|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2
+�
+E
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�����
+≤ 1
+2
+���hL(ℓ) − ℓ2E
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2αr⌋,1⌋
+���� ,
+32
+
+where we recall that σd = ℓd−α with α = 1/3. Using the triangle inequality we obtain
+2E
+�
+|D(1)
+2,τ|
+�
+≤ E
+������hL(ℓ) − ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=1
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������Fd
+⌊d2ατ⌋,1
+������
+�
+≤ E
+������hL(ℓ) − ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������Fd
+⌊d2ατ⌋,1
+������
+�
++ ℓ2E
+������1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��
+− 1 ∧ exp
+� d
+�
+i=1
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������
+�
+,
+where we used Jensen’s inequality to remove the conditional expectation in the last term. Recalling
+that x �→ 1 ∧ exp(x) is 1-Lipschitz, we can then bound the second term
+ℓ2E
+������1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��
+− 1 ∧ exp
+� d
+�
+i=1
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������
+�
+≤ ℓ2E
+����φd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+����
+�
+,
+(49)
+≤ ℓ2E
+�
+φd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2�1/2
+,
+where the final expectation converges to zero as d → ∞ by Proposition 17. For the remaining term
+in D(1)
+2,τ, since (Xd
+⌊d2ατ⌋,i, Zd
+⌊d2ατ⌋,i)2≤i≤n is independent of Fd
+⌊d2ατ⌋,1, we have
+ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������Fd
+⌊d2ατ⌋,1
+�
+= ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+���
+,
+and, using again the fact that x �→ 1 ∧ exp(x) is 1-Lipschitz, we have
+�����hL(ℓ) − ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��������
+≤
+�����hL(ℓ) − ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=1
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��������
++ ℓ2E
+����φd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+����
+�
+.
+The last term goes to 0 as shown in (49), and, as hL(ℓ) = ℓ2aL(ℓ), with
+aL(ℓ) = lim
+d→∞ E
+�
+1 ∧ exp
+� d
+�
+i=1
+φd,i
+��
+,
+33
+
+by Theorem 2, we obtain
+lim
+d→∞
+�����hL(ℓ) − ℓ2E
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+�������
+�
+= 0 ,
+showing that D(1)
+2,τ → 0 as d → ∞. To obtain convergence of T d
+1 , we observe that for any τ ∈
+[s, t], D(1)
+1,τ and D(1)
+2,τ follow the same distributions as D(1)
+1,s and D(1)
+2,s, since for any k ∈ N, Xd
+k has
+distribution πL
+d . Therefore, the convergence towards zero of E[|D(1)
+1,τ|] and E[|D(1)
+2,τ|] is uniform for
+τ ∈ [s, t], which gives us T d
+1 → 0 as d → ∞.
+Finally, consider T d
+2 . Using analogous arguments to those used for T d
+1 , we obtain
+E
+�
+|T d
+2 |
+�
+≤ C
+� t
+s
+ℓ2
+2 E
+�����E
+�
+bd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉
+����Fd
+⌊d2ατ⌋,1
+�
+− aL(ℓ, r)
+����
+�
+dτ
+≤ C
+� t
+s
+ℓ2
+2
+�
+E
+�
+|D(2)
+1,τ|
+�
++ E
+�
+|D(2)
+2,τ|
+�
+E
+�
+|D(2)
+3,τ|
+��
+dτ ,
+where we define
+D(2)
+1,τ = E
+��
+Zd
+⌈d2ατ⌉,1
+�2
+1Ad
+⌈d2ατ⌉
+����Fd
+⌊d2ατ⌋,1
+�
+− aL(ℓ, r) ,
+D(2)
+2,τ =
+�σd
+2 sgn(Xd
+⌊d2ατ⌋,1)1|Xd
+⌊d2ατ⌋,1|>σ2m
+d
+r/2 +
+1
+σ2m−1
+d
+rXd
+⌊d2ατ⌋,11|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2
+�2
+× E
+�
+1Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
+,
+D(2)
+3,τ = 2
+�σd
+2 sgn(Xd
+⌊d2ατ⌋,1)1|Xd
+⌊d2ατ⌋,1|>σ2m
+d
+r/2 +
+1
+σ2m−1
+d
+rXd
+⌊d2ατ⌋,11|Xd
+⌊d2ατ⌋,1|≤σ2m
+d
+r/2
+�
+× E
+�
+Zd
+⌈d2ατ⌉,11Ad
+⌈d2ατ⌉
+���Fd
+⌊d2ατ⌋,1
+�
+.
+Using (48), Cauchy-Schwarz’s inequality and the fact that the moments of Zd
+⌈d2ατ⌉,1 are bounded
+we have
+E
+�
+|D(2)
+2,τ|
+�
+≤ σ2
+d
+4
+−→
+d→∞ 0 ,
+E
+�
+|D(2)
+3,τ|
+�
+≤ Cσd −→
+d→∞ 0 ,
+since σd = ℓd−α with α = 1/3. The remaining term is bounded similarly to D(1)
+2,τ, using the fact
+that x �→ 1 ∧ exp(x) is 1-Lipschitz, we have
+E
+�
+|D(2)
+3,τ|
+�
+≤ E
+������E
+��
+Zd
+⌈d2ατ⌉,1
+�2
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��������Fd
+⌊d2ατ⌋,1
+�
+− aL(ℓ, r)
+�����
+�
++ E
+��
+Zd
+⌈d2ατ⌉,1
+�2 ���φd
+�
+Xd
+⌊d2ατ⌋,1, Zd
+⌈d2ατ⌉,1
+����
+�
+.
+34
+
+The second expectation is bounded as (49) using Cauchy-Schwarz’s inequality and Proposition 17.
+For the first expectation, we use the conditional independence of the components of Zd
+⌈d2ατ⌉ and
+write
+E
+��
+Zd
+⌈d2ατ⌉,1
+�2
+�
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+��������Fd
+⌊d2ατ⌋,1
+�
+= E
+��
+Zd
+⌈d2ατ⌉,1
+�2�
+E
+��
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+����
+= E
+��
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+����
+.
+It follows that E[|D(2)
+3,τ|] → 0 as d → ∞ since, by Theorem 2,
+�����E
+��
+1 ∧ exp
+� d
+�
+i=2
+φd
+�
+Xd
+⌊d2ατ⌋,i, Zd
+⌈d2ατ⌉,i
+����
+− aL(ℓ, r)
+����� → 0 .
+Combining the results for T d
+i , i = 1, . . . , 5 we obtain the result.
+Acknowledgments
+F.R.C. and G.O.R. acknowledge support from the EPSRC (grant # EP/R034710/1).
+G.O.R.
+acknowledges further support from the EPSRC (grant # EP/R018561/1) and the Alan Turing
+Institute. A.D. acknowledges support from the Lagrange Mathematics and Computing Research
+Center. The authors would like to thank ´Eric Moulines for helpful discussions.
+For the purpose of open access, the author has applied a Creative Commons Attribution (CC
+BY) licence to any Author Accepted Manuscript version arising from this submission.
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+A
+Proof of Theorem 1
+The proof of Theorem 1 follows that of [34, Theorem 1, Theorem 2] and consists of four propositions
+showing convergence of the log-acceptance probability to a normal random variable and (weak)
+convergence of the process (11) to a Langevin diffusion.
+We start by recalling and defining a number of quantities that we will use in the following proofs.
+Recall that σd = ℓ/dα, that λd = σ2m
+d r/2 where m ≥ 1/2 and r > 0 are to be chosen according to
+the different cases in Theorem 1. Recalling the expression of the proposal given in (9) and using
+the simplification given in (10), we define the proposal with starting point xd ∈ Rd,
+yd(xd, zd) = xd − σ2
+d
+2 ∇G
+�
+proxσ2m
+d
+r/2
+G
+(xd)
+�
++ σdzd ,
+where zd ∈ Rd. Since G is the d-times tensor product of g, the i-th component of the proposal only
+depends on the i-th components of xd and zd. Thus, we introduce the notation for any x, z ∈ R,
+yd(x, z) = x − σ2
+d
+2 g′ �
+proxσ2m
+d
+r/2
+g
+(x)
+�
++ σdz .
+With these notations, we obtain the proposal for the chain (Xd
+k)k≥0 using Y d
+k = yd(Xd
+k, Zd
+k+1) =
+(yd(Xd
+k,i, Zd
+k+1,i))i∈{1,...,d}.
+Let us define the generator of the discrete process (Xd
+k)k≥0 for all
+38
+
+V ∈ C∞
+c (Rd, R), i.e. infinitely differentiable R-valued multivariate functions with compact support,
+and any xd ∈ Rd,
+LdV (xd) = d2αE
+��
+V (yd(xd, Zd
+1)) − V (xd)
+� πd(yd(xd, Zd
+1))qd(yd(xd, Zd
+1), xd)
+πd(xd)qd(xd, yd(xd, Zd
+1))
+∧ 1
+�
+= d2αE
+�
+�
+V (yd(xd, Zd
+1)) − V (xd)
+�
+d
+�
+i=1
+exp
+�
+φd(xd
+i , Zd
+1,i)
+�
+∧ 1
+�
+,
+where the expectation is w.r.t. Zd
+1 = (Zd
+1,i)i∈{1,...,d}, a d-dimensional standard normal random
+variable, and where we defined
+φd(x, z) = log π(yd(x, z))q(yd(x, z), x)
+π(x)q(x, yd(x, z))
+(50)
+= g(yd(x, z)) − g(x) + log q(yd(x, z), x) − log q(x, yd(x, z)) .
+In the remainder we will work with one-dimensional functions V ∈ C∞
+c (R, R) applied to the first
+component of xd so that
+LdV (xd) = d2αE
+��
+V (yd(xd
+1, Zd
+1,1)) − V (xd
+1)
+� πd(yd(xd, Zd
+1))qd(yd(xd, Zd
+1), xd)
+πd(xd)qd(xd, yd(xd, Zd
+1))
+∧ 1
+�
+= d2αE
+�
+�
+V (yd(xd
+1, Zd
+1,1)) − V (xd
+1)
+�
+d
+�
+i=1
+exp
+�
+φd
+�
+xd
+i , Zd
+1,i
+��
+∧ 1
+�
+.
+(51)
+We also define �Ld to be a variant of Ld in which the first component of the acceptance ratio is
+omitted:
+�LdV (xd) = d2αE
+�
+�
+V (yd(xd
+1, Zd
+1,1)) − V (xd
+1)
+�
+d
+�
+i=2
+exp
+�
+φd
+�
+xd
+i , Zd
+1,i
+��
+∧ 1
+�
+.
+(52)
+We further define the generator of the Langevin diffusion (13)
+LV (x) = h(ℓ, r)
+2
+[V ′′(x) − g′(x)V ′(x)] ,
+(53)
+where h(ℓ, r) = ℓ2a(ℓ, r) is the speed of the diffusion and a(ℓ, r) = limd→∞ ad(ℓ, r) is given in
+Theorem 1.
+We will make use of the derivatives of g in (8) up to order 8, which we denote by g′, g′′, g′′′ and
+g(k) for all k = 4, . . . , 8.
+We recall that (gλ)′ is Lipschitz continuous with Lipschitz constant λ−1 [38, Proposition 12.19]
+and that (gλ)′(x) = λ−1(proxλ
+g(x)−x), hence proxλ
+g is Lipschitz continuous with Lipschitz constant
+1.
+A.1
+Auxiliary Results for the Proof of Case (a)
+First, we characterize the limit behaviour of the acceptance ratio (12).
+Proposition 4. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, then
+39
+
+(i) the log-acceptance ratio (50) satisfies
+φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) ,
+where C2(x, z) is given in (57), C3 and C4 are polynomials in z and the derivatives of g, such
+that E[C3(Xd
+0,1, Zd
+1,1)] = 0 and E[C2(Xd
+0,1, Zd
+1,1)2] = −2E[C4(Xd
+0,1, Zd
+1,1)];
+(ii) there exists sets Fd ⊆ Rd with d2απd(F c
+d) → 0 such that
+lim
+d→∞ sup
+xd∈Fd
+E
+������
+d
+�
+i=2
+φd(xd
+i , Zd
+1,i) − d−1/2
+d
+�
+i=2
+C2(xd
+i , Zd
+1,i) + ℓ4K1(r)2
+8
+�����
+�
+= 0 ,
+(54)
+where K1(r) is given in Theorem 1–(a).
+Proof. Take one component of the log-acceptance ratio
+φd(x, z) = g(yd(x, z)) − g(x) + log q(yd(x, z), x) − log q(x, yd(x, z)) ,
+with yd(x, z) = x−σ2
+dg′(proxσ2m
+d
+r/2
+g
+(x))/2+σdz. We have that φd(x, z) = R1(x, z, σd)+R2(x, z, σd),
+where
+R1(x, z, σ) = −g
+�
+x − σ2
+2 g′ �
+proxσ2mr/2
+g
+(x)
+�
++ σz
+�
++ g(x) ,
+R2(x, z, σ) = 1
+2z2 − 1
+2
+�
+z − σ
+2 g′
+�
+proxσ2mr/2
+g
+�
+x + σz − σ2
+2 g′ �
+proxσ2mr/2
+g
+(x)
+���
+−σ
+2 g′ �
+proxσ2mr/2
+g
+(x)
+��2
+.
+(55)
+Following the approach of [34] we approximate φd(x, z) with a Taylor expansion about σd → 0.
+(i) Using a Taylor expansion of order 5, we obtain
+φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) ,
+(56)
+where
+C2(x, z) = ℓ2
+2 (−rzg′′(x)g′(x)) ,
+(57)
+C3(x, z) and C4(x, z) are given in Appendix C.1.1 and we use the integral form for the re-
+mainder
+C5(x, z, σd) =
+� σd
+0
+∂5
+∂σ5 R(x, z, σ)
+����
+σ=u
+(σd − u)4
+4!
+du ,
+with u between 0 and σd and the derivatives of R1 and R2 given in Appendix C.2.
+In
+addition, integrating by parts and using the moments of Zd
+1,1 we find that E[C2(Xd
+0,1, Zd
+1,1)] =
+E[C3(Xd
+0,1, Zd
+1,1)] = 0 and
+2E
+�
+C4(Xd
+0,1, Zd
+1,1)
+�
++ E
+�
+C2(Xd
+0,1, Zd
+1,1)2�
+= 0 .
+40
+
+(ii) To construct the sets Fd, consider, for j = 3, 4, Fd,j = F 1
+d,j ∩ F 2
+d,j where we define
+F 1
+d,j =
+�
+xd ∈ Rd :
+�����
+d
+�
+i=2
+E
+�
+Cj(xd
+i , Zd
+1,i) − Cj(Xd
+0,i, Zd
+1,i)
+�
+����� ≤ d5/8
+�
+,
+and
+F 2
+d,j =
+�
+xd ∈ Rd :
+�����
+d
+�
+i=2
+Vj(xd
+i ) − E
+�
+Vj(Xd
+0,i)
+�
+����� ≤ d6/5
+�
+,
+where Vj(x) := Var(Cj(x, Zd
+1,1)).
+Using Markov’s inequality and the fact that Cj, Vj are
+bounded by polynomials since g and its derivatives are bounded by polynomials, it is easy to
+show that d1/2πd((F 1
+d,j)c) → 0 and d1/2πd((F 2
+d,j)c) → 0, from which follows d1/2πd(F c
+d,j) → 0
+as d → ∞. To prove L1 convergence of Cj for j = 3, 4, observe that
+E
+�
+�
+� d
+�
+i=2
+Cj(xd
+i , Zd
+1,i) − E
+�
+Cj(Xd
+0,1, Zd
+1,1)
+�
+�2�
+�
+=
+d
+�
+i=2
+Vj(xd
+i ) +
+� d
+�
+i=2
+E
+�
+Cj(xd
+i , Zd
+1,i) − Cj(Xd
+0,1, Zd
+1,1)
+�
+�2
+,
+and that, for xd ∈ Fd,j, we have
+E
+�
+�
+� d
+�
+i=2
+Cj(xd
+i , Zd
+1,i) − E
+�
+Cj(Xd
+0,1, Zd
+1,1)
+�
+�2�
+� ≤ E
+�
+Vj(xd
+1)
+�
+(d − 1) + d6/5 + d5/4 .
+Thus, the third and fourth term in the Taylor expansion (56) converge in L1 to 0 and
+−ℓ4K2
+1(r)/8 respectively. Now, consider C5(xd
+i , Zd
+1,i, σd). We can bound
+∂5R
+∂σ5 (x, z, σ) with
+the derivatives of g evaluated at
+x + σ2
+2 proxσ2mr/2
+g
+(x) + σz
+and
+proxσ2mr/2
+g
+(x) .
+Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that
+there exist polynomials p of the form
+A
+�
+1 +
+�
+proxσ2mr/2
+g
+(x)
+�N� �
+1 + zN� �
+1 + xN� �
+1 + σN�
+,
+for sufficiently large A and sufficiently large even integer N, such that
+����g(k)
+�
+x + σ2
+2 proxσ2mr/2
+g
+(x) + σdz
+����� ∨
+���g(k) �
+proxσ2mr/2
+g
+(x)
+����
+≤ p(proxσ2mr/2
+g
+(x), x, z, σd) .
+41
+
+In addition, | proxσ2mr/2
+g
+(x)| ≤ C(1 + |x|) for some C ≥ 1, and we can bound
+p(proxσ2mr/2
+g
+(x), x, z, σ) ≤ A
+�
+1 + zN� �
+1 + x2N� �
+1 + σN�
+.
+Therefore, we have
+E
+���C5(xd
+i , Zd
+1,i, σd)
+���
+≤ AE
+�
+1 + (Zd
+1,i)N� �
+1 + (xd
+i )2N� � σd
+0
+(1 + uN)(σd − u)4
+4!
+du
+≤ AE
+�
+1 + (Zd
+1,i)N� �
+1 + (xd
+i )2N�
+d−5/2
+≤ A
+�
+1 + (xd
+i )2N�
+d−5/2 ,
+where the last inequality follows since all the moments of Zd
+1 are bounded. Let us denote
+p(x) = A
+�
+1 + x2N�
+and
+Fd,5 =
+�
+xd ∈ Rd :
+�����d−1
+d
+�
+i=1
+p(xd
+i ) − E
+�
+p(Xd
+0,i)
+�
+����� < 1
+�
+.
+By Chebychev’s inequality we have πd(F c
+d,5) ≤ Var(p(Xd
+0,1))d−1. Additionally, for all xd ∈
+Fd,5,
+d
+�
+i=2
+E
+���C5(xd
+i , Zd
+1,i, σd)
+���
+≤
+d
+�
+i=2
+d−5/2 �
+E
+�
+p(Xd
+0,1)
+�
++ d−1�
+≤ d−3/2 �
+E
+�
+p(Xd
+0,1)
+�
++ 1
+�
+.
+Finally, set Fd = ∩5
+j=3Fd,j. On Fd the last three terms of (56) converge uniformly in L1,
+and (54) follows using the triangle inequality.
+Next, we compare the generator Ld and �Ld in (51) and (52) respectively.
+Proposition 5. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists sets Sd ⊆ Rd
+with d2απd(Sc
+d) → 0 such that for any V ∈ C∞
+c (R, R)
+lim
+d→∞ sup
+xd∈Sd
+���LdV (xd) − �LdV (xd)
+��� = 0 ,
+and
+lim
+d→∞ sup
+xd∈Sd
+E
+������
+�
+exp
+� d
+�
+i=1
+φd(xd
+i , Zd
+1,i)
+�
+∧ 1
+�
+−
+�
+exp
+� d
+�
+i=2
+φd(xd
+i , Zd
+1,i)
+�
+∧ 1
+������
+�
+= 0 .
+Proof. The function x �→ exp(x) ∧ 1 is Lipschitz continuous with Lipschitz constant 1, hence
+���LdV (xd) − �LdV (xd)
+��� ≤ d2αE
+���V
+�
+yd(xd
+1, Zd
+1,1)
+�
+− V (xd
+1)
+�� |R(xd
+1, Zd
+1,1, σd)|
+�
+,
+42
+
+where R(x, z, σ) = R1(x, z, σ) + R2(x, z, σ) as in (55). Using a Taylor expansion of order 1 about
+σ = 0 with integral remainder:
+R(x, z, σ) = R(x, z, 0) + ∂R
+∂σ (x, z, σ)
+����
+σ=0
+σ +
+� σ
+0
+∂2R
+∂σ2 (x, z, σ)
+����
+σ=u
+(σ − u)du ,
+we obtain
+R(x, z, σ) =
+� σ
+0
+∂2R
+∂σ2 (x, z, σ)
+����
+σ=u
+(σ − u)du ,
+where ∂2R
+∂σ2 (x, z, σ) is bounded by the derivatives of g evaluated at
+x + σ2
+2 proxσ2mr/2
+g
+(x) + σz
+and
+proxσ2mr/2
+g
+(x) .
+Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there
+exist polynomials p of the form
+A
+�
+1 +
+�
+proxσ2mr/2
+g
+(x)
+�N� �
+1 + zN� �
+1 + xN� �
+1 + σN�
+,
+for sufficiently large A and sufficiently large even integer N, such that
+����g(k)
+�
+x + σ2
+2 proxσ2mr/2
+g
+(x) + σz
+����� ∨
+���g(k) �
+proxσ2mr/2
+g
+(x)
+���� ≤ p(proxσ2mr/2
+g
+(x), x, z, σ) .
+Proceeding as in Proposition 4, we can bound
+p(proxσ2mr/2
+g
+(x), x, z, σ) ≤ A
+�
+1 + zN� �
+1 + x2N� �
+1 + σN�
+.
+Therefore, we have
+��R
+�
+xd
+1, Zd
+1,1, σd
+��� ≤ A
+�
+1 + (Zd
+1,1)N� �
+1 + (xd
+1)2N�
+×
+� σd
+0
+(1 + uN)(σd − u)du ≤ A
+�
+1 + (Zd
+1,1)N� �
+1 + (xd
+1)2N� σ2
+d
+2 .
+(58)
+Since V ∈ C∞
+c (R, R), there exists a constant C such that
+��V
+�
+yd(xd
+1, Zd
+1,1)
+�
+− V (xd
+1)
+�� ≤ C
+��yd(xd
+1, Zd
+1,1) − xd
+1
+��
+≤ C
+�
+σd|Zd
+1,1| + σ2
+d
+2
+���g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+1)
+����
+�
+.
+Recalling that g′(proxλ
+g(x)) = (gλ)′(x) with (gλ)′ Lipschitz continuous with Lipschitz constant λ−1,
+we have
+���g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+1)
+���� ≤
+2
+σ2m
+d r(1 + |xd
+1|) ,
+43
+
+and
+��V
+�
+yd(xd
+1, Zd
+1,1)
+�
+− V (xd
+1)
+�� ≤ C
+�
+σd|Zd
+1,1| + σ2−2m
+d
+r
+�
+1 + |xd
+1|
+��
+(59)
+≤ Cσd
+�
+|Zd
+1,1| + 1
+r
+�
+1 + |xd
+1|
+��
+,
+since m = 1/2. Combining (58) and (59) we obtain
+d2α ��V
+�
+yd(xd
+1, Zd
+1,1)
+�
+− V (xd
+1)
+�� ��R(xd
+1, Zd
+1,1, σd)
+��
+≤ Cσd
+�
+1 + (Zd
+1,1)N� �
+1 + (xd
+i )2N� �
+|Zd
+1,1| + 1
+r (1 + |xd
+1|)
+�
+,
+for some C > 0.
+Set Sd to be the set in which 1 + (xd
+1)2N+1 ≤ dα/2, applying Markov’s inequality we obtain
+d2απd(Sc
+d) = d2απd
+��
+1 + (xd
+1)2N+1�5 ≥ d5α/2�
+≤ d−α/2E
+��
+1 + (xd
+1)2N+1�5�
+−→
+d→∞ 0 .
+Recalling that |Zd
+1,1| and 1 + (Zd
+1,1)N are bounded, we have that
+sup
+xd∈Sd
+���LdV (xd) − �LdV (xd)
+��� ≤ Cdα/2 ℓ
+dα −→
+d→∞ 0 .
+The second results follows from (58) using the same argument.
+The following result considers the convergence to the generator of the Langevin diffusion (53).
+Proposition 6. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists sets Td ⊆ Rd
+with d2απd(T c
+d) → 0 as d → ∞, such that for any V ∈ C∞
+c (R, R)
+lim
+d→∞ sup
+xd∈Td
+����d2αE
+�
+V
+�
+yd(xd
+1, Zd
+1,1)
+�
+− V (xd
+1)
+�
+− ℓ2
+2 (V ′′(xd
+1) + g′(xd
+1)V ′(xd
+1))
+���� = 0 .
+Proof. Take
+yd(xd
+1, Zd
+1,1) = xd
+1 + σ2
+d
+2 g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+1)
+�
++ σdZd
+1,1 ,
+and use a Taylor expansion of order 2 of
+W(x, z, σ) = V
+�
+x + σ2
+2 g′ �
+proxσ2mr/2
+g
+(x)
+�
++ σz
+�
+,
+about σ = 0 with integral remainder:
+W(x, z, σ) = W(x, z, 0) + ∂W
+∂σ (x, z, σ)
+����
+σ=0
+σ + 1
+2
+∂2W
+∂σ2 (x, z, σ)
+����
+σ=0
+σ2
++
+� σ
+0
+∂3W
+∂σ3 (x, z, σ)
+����
+σ=u
+(σ − u)2
+2
+du .
+44
+
+Using the derivatives
+W(x, z, 0) = V (x) ,
+∂W
+∂σ (x, z, σ)
+����
+σ=0
+= V ′(x)z ,
+∂2W
+∂σ2 (x, z, σ) = V ′′(x)z2 + V ′(x)g′(x) ,
+and recalling that E
+�
+Zd
+1,1
+�
+= 0, E
+�
+(Zd
+1,1)2�
+= 1, we have
+E
+�
+V
+�
+yd(xd
+1, Zd
+1,1)
+�
+− V (xd
+1)
+�
+= σ2
+d
+2
+�
+V ′′(xd
+1) + V ′(xd
+1)g′(xd
+1)
+�
++ E
+�� σd
+0
+∂3W
+∂σ3 (xd
+1, Zd
+1,1, σ)
+����
+σ=u
+(σd − u)2
+2
+du
+�
+.
+Proceeding as in the previous proposition, we can bound
+����
+� σd
+0
+∂3W
+∂σ3 (xd
+1, Zd
+1,1, σ)
+��
+σ=u
+(σd − u)2
+2
+du
+���� ≤ A
+�
+1 + (Zd
+1,1)N� �
+1 + (xd
+i )2N�
+d−3α .
+Setting Td to be the set in which (1 + (xd
+1)2N) ≤ dα/2, the result follows by applying Markov’s
+inequality as in Proposition 5.
+Before proceeding to stating and proving the last auxiliary result, let us denote by ψ1 : R →
+[0, +∞) the characteristic function of the distribution N(0, K2
+1(r)), where K2
+1(r) is given in Theo-
+rem 1–(a),
+ψ1(t) = exp(−t2ℓ2K1(r)2/2) ,
+and by ψd
+1(xd; t) =
+�
+exp(itw)Qd
+1(xd; dw) the characteristic functions associated with the law
+Qd
+1(xd; ·) = L
+�
+d−1/2
+d
+�
+i=2
+C2(xd
+i , Zd
+1,i)
+�
+,
+Proposition 7. Under A0, A1 and A2, if α = 1/4, β = 1/8 and r > 0, there exists a sequence of
+sets Hd ⊆ Rd such that
+(i)
+lim
+d→∞ d2απd(Hc
+d) = 0 ,
+(ii) for all t ∈ R
+lim
+d→∞ sup
+xd∈Hd
+|ψd
+1(xd; t) − ψ1(t)| = 0 ,
+(iii)
+d−1/2
+d
+�
+i=2
+C2(xd
+i , Zd
+1,i)
+L
+−→ N
+�
+0, ℓ4K2
+1(r)
+2
+�
+,
+where
+L
+−→ denotes convergence in law,
+45
+
+(iv)
+lim
+d→∞ sup
+xd∈Hd
+�����E
+�
+1 ∧ exp
+�
+d−1/2
+d
+�
+i=2
+C2(xd
+i , Zd
+1,i) − ℓ4K2
+1(r)
+2
+��
+− 2Φ
+�
+−ℓ2K1(r)
+2
+������ ,
+where Φ is the distribution function of the standard normal random variable.
+Proof.
+(i) Define the functions hj(x) = [−g′′(x)g′(x)]j with j = 1, . . . , 4 and let Hd = Hd,1 ∩Hd,2
+where
+Hd,1 =
+�
+xd ∈ Rd :
+�����
+1
+d
+d
+�
+i=2
+hj(xd
+i ) −
+�
+R
+hj(u)π(u)du
+����� ≤ d1/3 for j = 1, . . . , 4
+�
+,
+Hd,2 =
+�
+xd ∈ Rd : |hj(xd
+i )| ≤ d2/3 for i = 1, . . . , d and j = 1, . . . , 4
+�
+.
+Using Chebychev’s inequality, the fact that the derivatives of g are bounded by polynomials
+and that π has finite moments, we have d1/2πd((Hd,1)c) → 0 as d → ∞. Similarly, by Markov’s
+inequality we have d1/2πd((Hd,2)c) → 0 as d → ∞.
+(ii) We follow [34, Lemma 3(b)] and decompose
+|ψd
+1(xd; t) − ψ1(t)| ≤
+�����ψd
+1(xd; t) −
+d
+�
+i=2
+�
+1 − t2
+2dv(xd
+i )
+������
+(60)
++
+�����
+d
+�
+i=2
+�
+1 − t2
+2dv(xd
+i )
+�
+−
+d
+�
+i=2
+exp
+�
+−t2 v(xd
+i )
+2d
+������
++
+�����
+d
+�
+i=2
+exp
+�
+−t2 v(xd
+i )
+2d
+�
+− exp
+�
+−t2 ℓ2K1(r)2
+2
+������
+where v(xd
+i ) = Var(C2(xd
+i , Zd
+1,i)) = ℓ4E[C2(xd
+i , Zd
+1,i)2].
+For the first term, decompose the
+characteristic function ψd
+1(xd; t) = �d
+i=2 θd
+i (xd
+i ; t) as the product of the characteristic functions
+of d−1/2Wi where we define Wi = C2(xd
+i , Zd
+1,i), using [15, equation (3.3.3)] as in the proof of
+[15, Theorem 3.4.10] we obtain
+����θd
+i (xd
+i ; t) −
+�
+1 − t2
+2dv(xd
+i )
+����� ≤ EZ
+� |t|3
+d3/2
+|Wi|3
+3!
+∧ 2|t|2
+d
+|Wi|2
+2!
+�
+≤ EZ
+� |t|3
+d3/2
+|Wi|3
+3!
+; |Wi| ≤ d1/2ε
+�
++ t2
+d EZ
+�
+|Wi|2; |Wi| > d1/2ε
+�
+≤ ε|t|3
+6d EZ
+�
+|Wi|2�
++
+t2
+ε2d2 EZ
+�
+|Wi|4�
+for any ε > 0. For sufficiently large d, we have that t2v(xd
+i )/(2d) ≤ 1 for x ∈ Hd,2, and we
+can use [15, Lemma 3.4.3]
+�����ψd
+j (xd; t) −
+d
+�
+i=2
+�
+1 − t2
+2dv(xd
+i )
+������ ≤
+d
+�
+i=2
+�ε|t|3
+6d EZ
+�
+|Wi|2�
++
+t2
+ε2d2 EZ
+�
+|Wi|4��
+≤ εℓ4|t|3
+6
+(K2
+1(r) + D1d−1/3) + t2
+ε2d(E
+�
+|Wi|4�
++ D2ℓ8d−1/4)
+46
+
+where the last inequality follows from the fact that xd ∈ Hd,2 and D1, D2 are positive con-
+stants. For any δ > 0 we can chose ε small enough so that the first term in the above is less
+than δ/2 and we can chose d sufficiently large to make the second term less than δ/2. Thus,
+for any δ > 0 we can find ε > 0 and d ∈ N such that
+�����ψd
+1(xd; t) −
+d
+�
+i=2
+�
+1 − t2
+2dv(xd
+i )
+������ < δ;
+the uniform convergence then follows. The second term in (60) converges to 0 uniformly for
+all xd ∈ Hd,1; while for the third term in (60) we use again [15, Lemma 3.4.3]
+�����
+d
+�
+i=2
+exp
+�
+−t2 v(xd
+i )
+2d
+�
+− exp
+�
+−t2 ℓ2K1(r)2
+2
+������ ≤
+d
+�
+i=2
+t4v(xd
+i )2
+4d2
+→ 0
+for all x ∈ Hd,2. The result then follows.
+(iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.g. [41,
+Theorem 1, page 322]).
+(iv) This statement follows directly from (iii) and [33, Proposition 2.4].
+A.2
+Auxiliary Results for the Proof of Case (b)
+First, we characterize the limit behaviour of the acceptance ratio (12). The following result is an
+extension of [34, Lemma 1].
+Proposition 8. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, then
+(i) the the log-acceptance ratio (50) satisfies
+φd(xd
+i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi)
++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd),
+where C3(xi, Zi) is given in (61), C4, C5, C6 are polynomials in Zi and the derivatives of g,
+such that EX [EZ [Cj(X, Z)]] = 0 for j = 3, 4, 5 and
+EX
+�
+EZ
+�
+C3(X, Z)2��
+= −2EX [EZ [C6(X, Z)]] ;
+(ii) there exists sets Fd ⊆ Rd with d2απd(F c
+d) → 0 such that
+lim
+d→∞ sup
+xd∈Fd
+E
+������
+d
+�
+i=2
+φd(xd
+i , Yi) − d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi) + ℓ6K2(r)2
+2
+�����
+�
+= 0,
+where K2(r) is given in Theorem 1–(b).
+47
+
+Proof. Take one component of the log-acceptance ratio
+φd(xd
+i , Yi) = g(Yi) − g(xd
+i ) + log q(Yi, xd
+i ) − log q(xd
+i , Yi)
+with Yi = xd
+i − σ2
+d
+2 g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+i )
+�
++ σdZi and Zi ∼ N(0, 1). Proceeding as in the proof for
+case (a), we have that φd(xd
+i , Yi) = R1(xd
+i , Z, σd) + R2(xd
+i , Z, σd) where R1, R2 are given in (55).
+Following the approach of [34] we approximate φd(xd
+i , Yi) with a Taylor expansion about σd = 0.
+(i) Using a Taylor expansion of order 7, we find that
+φd(xd
+i , Yi) = d−1/2C3(xd
+i , Zi) + d−2/3C4(xd
+i , Zi) + d−5/6C5(xd
+i , Zi)
++ d−1C6(xd
+i , Zi) + C7(xd
+i , Zi, σd),
+where
+C3(x, z) = ℓ3
+6
+�1
+2g′′′(x)z3 − 3
+2g′′(x)g′(x)z (1 + 2r)
+�
+,
+(61)
+C4(x, z), C5(x, z) and C6(x, z) are given in Appendix C.1.2 and integral form of the remainder
+C7(xd
+i , Zi, σd) =
+� σd
+0
+∂7
+∂σ7 R(xd
+i , Zi, σ)|σ=ϵ
+(σd − ϵ)6
+6!
+dϵ,
+with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2. In
+addition, integrating by parts and using the moments of Z we find that EX [EZ [C3(X, Z)]] =
+EX [EZ [C4(X, Z)]] = EX [EZ [C5(X, Z)]] = 0 and
+EX [EZ [C6(X, Z)]]
+= ℓ6
+�
+− 1
+16
+�
+r + 2r2�
+[g′′(X)g′(X)]2 − 1
+16
+�1
+2 + r
+�
+g′′(X)3 − 5
+96g′′′(X)2
+�
+,
+which shows that EX
+�
+EZ
+�
+C3(X, Z)2 + 2C6(X, Z)
+��
+= 0.
+(ii) The proof of this result follows using the same steps as case (a) and is analogous to that of
+[34, Lemma 1].
+Next, we compare the generator Ld and �Ld in (51) and (52) respectively, extending [34, Theorem
+3].
+Proposition 9. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists sets
+Sd ⊆ Rd with d2απd(Sc
+d) → 0 such that for any V ∈ C∞
+c (R, R)
+lim
+d→∞ sup
+xd∈Sd
+���LdV (xd) − �LdV (xd)
+��� = 0
+and
+lim
+d→∞ sup
+xd∈Sd
+E
+������
+πd(Y )qd(Y , xd)
+πd(xd)qd(xd, Y ) ∧ 1 −
+d
+�
+i=2
+φd(xd
+i , yi) ∧ 1
+�����
+�
+= 0.
+48
+
+Proof. The function x �→ exp(x) ∧ 1 is Lipschitz continuous with Lipschitz constant 1, hence
+���LdV (xd) − �LdV (xd)
+��� ≤ d2αE
+�
+|V (Y1) − V (xd
+1)||R(xd
+1, Z1, σd)|
+�
+where R(x, z, σ) = R1(x, z, σ) + R2(x, z, σ) as in (55). Using a Taylor expansion of order 1 about
+σ = 0 with integral remainder:
+R(x, z, σ) = R(x, z, 0) + ∂R
+∂σ (x, z, σ)|σ=0σ +
+� σ
+0
+∂2R
+∂σ2 (x, z, σ)|σ=ϵ(σ − ϵ)dϵ,
+we obtain
+R(xd
+1, Z, σd) =
+� σd
+0
+∂2R
+∂σ2 (x, z, σ)|σ=ϵ(σd − ϵ)dϵ,
+where ∂2R
+∂σ2 (x, z, σ) is bounded by the derivatives of g evaluated at
+x + σ2
+d
+2 proxσ2mr/2
+g
+(x) + σz
+and
+proxσ2mr/2
+g
+(x).
+Under our assumptions, the derivatives of g are bounded by polynomials M0, it follows that there
+exist polynomials p of the form A
+�
+1 +
+�
+proxσ2mr/2
+g
+(x)
+�N� �
+1 + zN� �
+1 + xN� �
+1 + σN�
+, for suffi-
+ciently large A and sufficiently large even integer N, such that
+����g(k)
+�
+x + σ2
+2 proxσ2mr/2
+g
+(x) + σz
+����� ∨
+���g(k) �
+proxσ2mr/2
+g
+(x)
+���� ≤ p(proxσ2mr/2
+g
+(x), x, z, σ).
+Proceeding as in Proposition 8, we can bound
+p(proxσ2mr/2
+g
+(x), x, z, σ) ≤ A
+�
+1 + zN� �
+1 + x2N� �
+1 + σN�
+.
+Therefore, we have
+|R(xd
+1, Z1, σd)| ≤ A
+�
+1 + ZN
+1
+� �
+1 + (xd
+i )2N�
+×
+� σd
+0
+(1 + ϵN)(σd − ϵ)dϵ ≤ A
+�
+1 + ZN
+1
+� �
+1 + (xd
+1)2N� σ2
+d
+2 .
+Since V ∈ C∞
+c (R, R), there exists a constant C such that
+|V (Y1) − V (xd
+1)| ≤ C|Y1 − xd
+1| ≤ C
+�
+σd|Z1| + σ2
+d
+2 |g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+1)
+�
+|
+�
+.
+Under A3, g′ is Lipschitz continuous and we have, for some C ≥ 1,
+|g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+1)
+�
+| ≤ C(1 + | proxσ2m
+d
+r/2
+g
+(xd
+1)|) ≤ C(1 + |xd
+1|),
+where we used the fact that proxλ
+g is 1-Lipschitz continuous for all λ > 0. The result then follows
+exactly as in [34, Theorem 3].
+49
+
+The following result considers the convergence to the generator of the Langevin diffusion (53)
+and is a generalization of [34, Lemma 2].
+Proposition 10. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists sets
+Td ⊆ Rd with d2απd(T c
+d) → 0 such that for any V ∈ C∞
+c (R, R)
+lim
+d→∞ sup
+xd∈Td
+����d2αE
+�
+V (Y1) − V (xd
+1)
+�
+− ℓ2
+2 (V ′′(xd
+1) + g′(xd
+1)V ′(xd
+1))
+���� = 0.
+Proof. The proof is identical to that of Proposition 6.
+Before proceeding to stating and proving the last auxiliary result, let us denote by ψ2 : R →
+[0, +∞) the characteristic function of the distribution N(0, K2
+2(r)), where K2
+2(r) is given in Theo-
+rem 1,
+ψ2(t) = exp(−t3ℓ2K2(r)2/2),
+and by ψd
+2(xd; t) =
+�
+exp(itw)Qd
+2(xd; dw) the characteristic functions associated with the law
+Qd
+2(xd; ·) = L
+�
+d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi)
+�
+.
+The following result extends [34, Lemma 3].
+Proposition 11. Under A0, A1, A2 and A3, if α = 1/6, β = 1/6 and r > 0, there exists a
+sequence of sets Hd ⊆ Rd such that
+(i)
+lim
+d→∞ d2απd(Hc
+d) = 0,
+(ii) for all t ∈ R
+lim
+d→∞ sup
+xd∈Hd
+|ψd
+2(xd; t) − ψ2(t)| = 0,
+(iii)
+d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi)
+L
+−→ N
+�
+0, ℓ6K2
+2(r)
+2
+�
+,
+where
+L
+−→ denotes convergence in law,
+(iv)
+lim
+d→∞ sup
+xd∈Hd
+�����EZ
+�
+1 ∧ exp
+�
+d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi) − ℓ6K2
+2(r)
+2
+��
+− 2Φ
+�
+−ℓ3K2(r)
+2
+������ .
+50
+
+Proof.
+(i) The proof is analogous to that of Proposition 7 and follows the same steps of that of
+[34, Lemma 3(a)].
+(ii) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34,
+Lemma 3(b)].
+(iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.g. [41,
+Theorem 1, page 322]).
+(iv) This statement follows directly from (iii) and [33, Proposition 2.4].
+A.3
+Auxiliary Results for the Proof of Case (c)
+First, we characterize the limit behaviour of the acceptance ratio (12).
+Proposition 12. Under A0, A1, A2 and A3 and, if α = 1/6, β = m/6 for m > 1 and r > 0,
+then
+(i) the log-acceptance ratio (50) satisfies
+φd(xd
+i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi)
++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd),
+where C3(xi, Zi) is given in (62), C4, C5, C6 are polynomials in Zi and the derivatives of g,
+such that EX [EZ [Cj(X, Z)]] = 0 for j = 3, 4, 5 and
+EX
+�
+EZ
+�
+C3(X, Z)2��
+= −2EX [EZ [C6(X, Z)]] ;
+(ii) there exists sets Fd ⊆ Rd with d2απd(F c
+d) → 0 such that
+lim
+d→∞ sup
+xd∈Fd
+E
+������
+d
+�
+i=2
+φd(xd
+i , Yi) − d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi) + ℓ6K2
+3
+2
+�����
+�
+= 0,
+where K3 = K2(0).
+Proof. Take one component of the log-acceptance ratio
+φd(xd
+i , Yi) = g(Yi) − g(xd
+i ) + log q(Yi, xd
+i ) − log q(xd
+i , Yi)
+with Yi = xd
+i − σ2
+d
+2 g′ �
+proxσ2m
+d
+r/2
+g
+(xd
+i )
+�
++ σdZi and Zi ∼ N(0, 1). Proceeding as in the proof for
+case (a), we have that φd(xd
+i , Yi) = R1(xd
+i , Z, σd) + R2(xd
+i , Z, σd) where R1, R2 are given in (55).
+Following the approach of [34] we approximate φd(xd
+i , Yi) with a Taylor expansion about σd = 0.
+(i) Using a Taylor expansion of order 7, we find that
+φd(xd
+i , Yi) = d−1/2C3(xd
+i , Zi) + d−2/3C4(xd
+i , Zi) + d−5/6C5(xd
+i , Zi)
++ d−1C6(xd
+i , Zi) + C7(xd
+i , Zi, σd),
+51
+
+where
+C3(x, z) = ℓ3
+6
+�1
+2g′′′(x)z3 − 3
+2g′′(x)g′(x)z
+�
+,
+(62)
+C4(x, z), C5(x, z) and C6(x, z) are given in Appendix C.1.3 and integral form of the remainder
+C7(xd
+i , Zi, σd) =
+� σd
+0
+∂7
+∂σ7 R(xd
+i , Zi, σ)|σ=ϵ
+(σd − ϵ)6
+6!
+dϵ,
+with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2. In
+addition, integrating by parts and using the moments of Z we find that EX [EZ [C3(X, Z)]] =
+EX [EZ [C4(X, Z)]] = EX [EZ [C5(X, Z)]] = 0 and
+EX [EZ [C6(X, Z)]] = ℓ6
+�
+− 1
+32g′′(X)3 − 5
+96g′′′(X)2
+�
+,
+which shows that EX
+�
+EZ
+�
+C3(X, Z)2 + 2C6(X, Z)
+��
+= 0.
+(ii) The proof of this result follows using the same steps as case (a) and is analogous to that of
+[34, Lemma 1].
+Next, we compare the generator Ld and �Ld in (51) and (52) respectively.
+Proposition 13. Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there
+exists sets Sd ⊆ Rd with d2απd(Sc
+d) → 0 such that for any V ∈ C∞
+c (R, R)
+lim
+d→∞ sup
+xd∈Sd
+���LdV (xd) − �LdV (xd)
+��� = 0
+and
+lim
+d→∞ sup
+xd∈Sd
+E
+������
+πd(Y )qd(Y , xd)
+πd(xd)qd(xd, Y ) ∧ 1 −
+d
+�
+i=2
+φd(xd
+i , yi) ∧ 1
+�����
+�
+= 0.
+Proof. The proof is identical to that of Proposition 9.
+The following result considers the convergence to the generator of the Langevin diffusion (53).
+Proposition 14. Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there
+exists sets Td ⊆ Rd with d2απd(T c
+d) → 0 such that for any V ∈ C∞
+c (R, R)
+lim
+d→∞ sup
+xd∈Td
+����d2αE
+�
+V (Y1) − V (xd
+1)
+�
+− ℓ2
+2 (V ′′(xd
+1) + g′(xd
+1)V ′(xd
+1))
+���� = 0.
+Proof. The proof is identical to that of Proposition 6.
+52
+
+Before proceeding to stating and proving the last auxiliary result, let us denote by ψ3 : R →
+[0, +∞) the characteristic function of the distribution N(0, K2
+3), where K2
+3 = K2
+2(0),
+ψ3(t) = exp(−t2ℓ3K2
+3/2),
+and by ψd
+3(xd; t) =
+�
+exp(itw)Qd
+3(xd; dw) the characteristic functions associated with the law
+Qd
+3(xd; ·) = L
+�
+d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi)
+�
+.
+Proposition 15. Under A0, A1, A2 and A3, if α = 1/6, β = m/6 for m > 1 and r > 0, there
+exists a sequence of sets Hd ⊆ Rd such that
+(i)
+lim
+d→∞ d2απd(Hc
+d) = 0,
+(ii) for all t ∈ R
+lim
+d→∞ sup
+xd∈Hd
+|ψd
+3(xd; t) − ψ3(t)| = 0,
+(iii)
+d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi)
+L
+−→ N
+�
+0, ℓ6K2
+3
+2
+�
+,
+where
+L
+−→ denotes convergence in law,
+(iv)
+lim
+d→∞ sup
+xd∈Hd
+�����EZ
+�
+1 ∧ exp
+�
+d−1/2
+d
+�
+i=2
+C3(xd
+i , Zi) − ℓ6K2
+3
+2
+��
+− 2Φ
+�
+−ℓ3K3
+2
+������ .
+Proof.
+(i) The proof is analogous to that of Proposition 7 and follows the same steps of that of
+[34, Lemma 3(a)].
+(ii) The proof is analogous to that of Proposition 7 and follows the same steps of that of [34,
+Lemma 3(b)].
+(iii) This is a straightforward consequence of (ii) and the L´evy’s continuity Theorem (e.g. [41,
+Theorem 1, page 322]).
+(iv) This statement follows directly from (iii) and [33, Proposition 2.4].
+53
+
+A.4
+Proof of Theorem 1
+Proof of Theorem 1.
+(a) The asymptotic acceptance rate follows by combining Propositions 4–
+6 with part (iv) of Proposition 7 as in the proof of [34, Theorem 1].
+To prove the weak
+convergence of the process it suffices to show that there exists events F ⋆
+d ∈ Rd such that for
+all t > 0
+P
+�
+Ld
+t ∈ F ⋆
+d for all 0 ≤ s ≤ t
+�
+→ 1
+and
+lim
+d→∞ sup
+xd∈F ⋆
+d
+��LdV (xd) − LV (xd)
+��
+for all V ∈ C∞
+c (R, R) [17, Chapter 4, Corollary 8.7]. We take F ⋆
+d = Fd ∩ Sd ∩ Td ∩ Hd. Then,
+d2απd ((F ⋆
+d )c) → 0 and
+P
+�
+Ld
+t ∈ F ⋆
+d for all 0 ≤ s ≤ t
+�
+→ 1
+for all fixed t. Combining Propositions 4–7 we obtain convergence of the generators.
+To obtain the value of a(ℓ, r) maximizing the speed, we observe that K2
+1(r) is a function of
+the ratio r = c2/ℓ2m = c2/ℓ only, we can take c ∝ ℓ1/2 so that K2
+1(r) is constant for given
+c. Using the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is maximized at the
+unique value of ℓ such that a(ℓ, r) = 0.452.
+(b) The proof is analogous to that of case (a) replacing Propositions 4, 5, 6 and 7 with Proposi-
+tions 8, 9, 10 and 11. To obtain the value of a(ℓ, r) maximizing the speed, we observe that
+K2
+2(r) is a function of the ratio r = c2/ℓ2m = c2/ℓ2 only, we can take c ∝ ℓ so that K2
+2(r) is
+constant for given c. Using the same substitution as in [34, Theorem 2] we find that h(ℓ, r) is
+maximized at the unique value of ℓ such that a(ℓ, r) = 0.574.
+(c) The proof is analogous to that of case (a) replacing Propositions 4, 5, 6 and 7 with Proposi-
+tions 12, 13, 14 and 15. To obtain the value of a(ℓ, r) maximizing the speed, we observe that
+K2
+3 is constant w.r.t. r, we can use the same substitution as in [34, Theorem 2] we find that
+h(ℓ, r) is maximized at the unique value of ℓ such that a(ℓ, r) = 0.574.
+B
+Numerical Experiments
+B.1
+Differentiable Targets
+We collect here a number of numerical experiments confirming the results in Section 3.2. To do so,
+we consider the Gaussian distribution in Example 1 and four algorithmic settings summarized in
+Table 1 which correspond to the three cases identified in Theorem 1 and MALA.
+The first plot in Figure 2–5 show that for values of α different from those identified in Theorem 1
+the acceptance ratio ad(ℓ, r) becomes degenerate as d increases. For the values of α identified in
+Theorem 1 we analyze the influence of ℓ on the acceptance ad(ℓ, r) (second plot), obtaining for
+d → +∞ the expression given by Theorem 1–(a) for Figure 2, the expression in Theorem 1–(b) for
+Figures 3 and 5 and that in Theorem 1–(c) for Figure 4.
+54
+
+Case
+Figure
+Algorithm
+α
+β
+m
+r
+(a)
+2
+Proximal MALA
+1/4
+1/8
+1/2
+1
+(b)
+3
+P-MALA
+1/6
+1/6
+1
+1
+(c)
+4
+Proximal MALA
+1/6
+1/2
+3
+2
+—
+5
+MALA
+1/6
+1/6
+1
+≈ 0
+Table 1: Algorithm setting for the simulation study on the Gaussian target.
+Finally, we consider the relationship between acceptance ratio ad(ℓ, r) and the speed of the
+diffusion h(ℓ, r) approximated by the expected squared jumping distance (see, e.g. [18])
+ESJDd := d2αE
+�
+(Xd
+0 − Xd
+1)2�
+.
+Looking at the last plot in Figure 2–5 we find that, even for relatively small values of d, the shape of
+the plot of ESJDd as a function of the acceptance ad(ℓ, r) is similar to that of the theoretical limit.
+This suggests that tuning the acceptance ratio to be approximately 0.452 when α = 1/4, β = 1/8
+and approximately 0.574 when α = 1/6, β = m/6 with m ≥ 1 should generally guarantee high
+efficiency.
+B.2
+Laplace Target
+We collect here a number of numerical experiments confirming the results for the Laplace distribu-
+tion in Section 3.2. Similarly to Appendix B.1 we consider three algorithmic settings, summarized
+in Table 2.
+Figure
+Algorithm
+α
+β
+m
+r
+6
+MALA
+1/3
+1/3
+1
+≈ 0
+7
+P-MALA
+1/3
+1/3
+1
+1
+8
+Proximal MALA
+1/3
+1
+3
+2
+Table 2: Algorithm setting for the simulation study on the Laplace target.
+The first plot in Figures 6–8 shows that for values of α ̸= 1/3 the acceptance ration ad(ℓ, r)
+becomes degenerate as d increases; while the second plot shows that, for sufficiently large d, the
+average acceptance ratio and the ESJDd converge to aL(ℓ) and hL(ℓ) given in Theorems 2 and 3,
+respectively. In the case m = 3, r = 2 in Figure 8, we find that the behaviour for low values of
+d significantly differs from the limiting one. For values of d lower than 130 the ESJDd achieves
+its maximum at a value of the acceptance ad(ℓ, r) different from that predicted by Theorem 3. In
+practice, this might mean that for low dimensional settings the recommended choice of a(ℓ, c) =
+0.360 is far from optimal. Similar behaviours for small d have also been observed in the case of
+RWM and MALA (e.g., [40, Section 2.1]).
+55
+
+0
+10
+20
+30
+40
+50
+60
+70
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 1
+8
+= 1
+4
+= 1
+2
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 15
+d = 50
+d = 300
+d = 10000
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+d=15
+d=50
+d=300
+d=10000
+theory
+ad(ℓ, r)
+ESJDd
+Figure 2: Case (a): Proximal MALA with Gaussian target and m = 1/2, r = 1. Average ac-
+ceptance rate for different choices of α (first); acceptance rate as a function of ℓ for increasing
+dimension d (second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+56
+
+10
+20
+30
+40
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 1
+3
+= 1
+6
+= 1
+12
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 3
+d = 41
+d = 300
+d = 10000
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d=41
+d=300
+d=30000
+theory
+ad(ℓ, r)
+ESJDd
+Figure 3: Case (b): Proximal MALA with Gaussian target and m = 1, r = 1 (P-MALA). Average
+acceptance rate for different choices of α (first); acceptance rate as a function of ℓ for increasing
+dimension d (second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+57
+
+10
+20
+30
+40
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 1
+3
+= 1
+6
+= 1
+12
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 100
+d = 1000
+d = 60000
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+d=100
+d=1000
+d=60000
+theory
+ad(ℓ, r)
+ESJDd
+Figure 4: Case (c): Proximal MALA with Gaussian target and m = 3, r = 2. Average acceptance
+rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d
+(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+58
+
+10
+20
+30
+40
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 1
+3
+= 1
+6
+= 1
+12
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 3
+d = 22
+d = 41
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+d=22
+d=100
+d=5000
+theory
+ad(ℓ, r)
+ESJDd
+Figure 5: Proximal MALA with Gaussian target and m = 1, r → 0 (MALA). Average acceptance
+rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d
+(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+59
+
+10
+20
+30
+40
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 2
+3
+= 1
+3
+= 1
+6
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 5
+d = 26
+d = 100
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+d = 15
+d = 80
+d = 1000
+theoretical limit
+ad(ℓ, r)
+ESJDd
+Figure 6: Proximal MALA with Laplace target and m = 1, r = 0 (sG-MALA). Average acceptance
+rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d
+(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+60
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 2
+3
+= 1
+3
+= 1
+6
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 5
+d = 50
+d = 300
+d = 10000
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+d = 300
+d = 1000
+d = 15000
+theoretical limit
+ad(ℓ, r)
+ESJDd
+Figure 7: Proximal MALA with Laplace target and m = 1, r = 1 (P-MALA). Average acceptance
+rate for different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d
+(second); ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+61
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+= 2
+3
+= 1
+3
+= 1
+6
+d
+ad(ℓ, r)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d = 15
+d = 32
+d = 58
+d = 300
+theoretical limit
+ℓ
+ad(ℓ, r)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+d = 15
+d = 50
+d = 150
+d = 1000
+theoretical limit
+ad(ℓ, r)
+ESJDd
+Figure 8: Proximal MALA with Laplace target and m = 3, r = 2. Average acceptance rate for
+different choices of α (first); acceptance rate as a function of ℓ for increasing dimension d (second);
+ESJDd as a function of the acceptance rate ad(ℓ, r) (third).
+62
+
+C
+Taylor Expansions for the Results on Differentiable Tar-
+gets
+C.1
+Coefficients of the Taylor Expansion
+We collect here the coefficients of the Taylor expansions in Propositions 4, 8 and 12.
+C.1.1
+Case (a)
+If α = 1/4, β = 1/8 and r > 0, then the log-acceptance ratio (50) satisfies
+φd(x, z) = d−1/2C2(x, z) + d−3/4C3(x, z) + d−1C4(x, z) + C5(x, z, σd) ,
+where
+C2(x, z) = ℓ2
+2 (−rzg′′(x)g′(x)) ,
+and
+C3(x, z) = ℓ3
+6
+�z3
+2 g′′′(x) − 3
+2z2rg′(x)g′′′(x) − 3
+2rz2 [g′′(x)]2 + 3
+4zr2g′′′(x) [g′(x)]2
+−3
+2zg′(x)g′′(x) + 3
+2r [g′(x)]2 g′′(x)
+�
+,
+C4(x, z) = ℓ4
+24
+�
+g(4)(x)
+�
+z4 − zr3
+2 [g′(x)]3 − 3z3rg′(x) + 3
+2z2r2 [g′(x)]2
+�
++ g′′′(x)
+�
+−6z2g′(x) − 9rz3g′′(x) + 9z2r2g′(x)g′′(x) + 9rz [g′(x)]2
+−9
+2zr3 [g′(x)]2 g′′(x) − 3
+2r2 [g′(x)]3
+�
++ 12rzg′(x) [g′′(x)]2 + 3g′′(x) [g′(x)]2 − 3z2 [g′′(x)]2 + 3z2r2 [g′′(x)]3
+−6r2 [g′(x)g′′(x)]2 − 3zr3g′(x) [g′′(x)]3�
+,
+and we use the integral form for the remainder
+C5(x, z, σd) =
+� σd
+0
+∂5
+∂σ5 R(x, z, σ)
+����
+σ=u
+(σd − u)4
+4!
+du ,
+with u between 0 and σd and the derivatives of R1 and R2 given in Appendix C.2.
+C.1.2
+Case (b)
+If α = 1/6, β = 1/6 and r > 0, the the log-acceptance ratio (50) satisfies
+φd(xd
+i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi)
++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd),
+63
+
+where
+C3(x, z) = ℓ3
+6
+�1
+2g′′′(x)z3 − 3
+2g′′(x)g′(x)z (1 + 2r)
+�
+,
+and
+C4(x, z) = ℓ4
+24
+�
+z4g(4)(x) − 6z2g′′′(x)g′(x)(1 + r)
+−3z2 [g′′(x)]2 (1 + 2r) + 3g′′(x) [g′(x)]2 (1 + 2r)
+�
+,
+C5(x, z) = ℓ5
+120
+�3
+2z5g(5)(x) − 15z3g(4)(x)g′(x)(1 + r) + 15z [g′(x)]2 g′′′(x)
+�3
+2 + 3r + r2
+�
++15z(1 + 4r + 2r2)g′(x) [g′′(x)]2 − 15z3g′′(x)g′′′(x)(1 + 3r)
+�
+,
+C6(x, z) = ℓ6
+720
+�
+2z6g(6)(x) − 30 (1 + r) z4g′(x)g(5)(x) + 45
+�
+2 + 4r + r2�
+z2 [g′(x)]2 g(4)(x)
++ 90
+�
+r + r2�
+z2 [g′′(x)]3 − 15
+�
+2 + 6r + 3r2�
+g′′′(x) [g′(x)]3
+− 30 (1 + 4r) z4g′′(x)g(4)(x) + 45
+�
+3 + 16r + 6r2�
+z2g′(x)g′′(x)g′′′(x)
+−45
+2 (1 + 4r) z4 [g′′′(x)]2 − 45
+2
+�
+1 + 8r + 8r2�
+[g′(x)g′′(x)]2
+�
+,
+and integral form of the remainder
+C7(xd
+i , Zi, σd) =
+� σd
+0
+∂7
+∂σ7 R(xd
+i , Zi, σ)|σ=ϵ
+(σd − ϵ)6
+6!
+dϵ,
+with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2.
+C.1.3
+Case (c)
+If α = 1/6, β = m/6 for m > 1 and r > 0, then the log-acceptance ratio (50) satisfies
+φd(xd
+i , Yi) = d−1/2C3(xi, Zi) + d−2/3C4(xi, Zi)
++ d−5/6C5(xi, Zi) + d−1C6(xi, Zi) + C7(xi, Zi, σd),
+where
+C3(x, z) = ℓ3
+6
+�1
+2g′′′(x)z3 − 3
+2g′′(x)g′(x)z
+�
+64
+
+and
+C4(x, z) = ℓ4
+24
+�
+�
+�
+�
+�
+�
+�
+�
+�
+z4g(4)(x) − 6z2g′′′(x)g′(x) − 3z2 [g′′(x)]2 + 3g′′(x) [g′(x)]2
+−12zrg′(x)g′′(x)
+if m = 3/2
+z4g(4)(x) − 6z2g′′′(x)g′(x) − 3z2 [g′′(x)]2 + 3g′′(x) [g′(x)]2
+otherwise
+,
+C5(x, z) = ℓ5
+120
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+3
+2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45
+2 z [g′(x)]2 g′′′(x) + 15zg′(x) [g′′(x)]2
+−15z3g′′(x)g′′′(x) − 30z2r [g′′(x)]2 + 30r [g′(x)]2 g′′(x)
+−30rz2g′(x)g′′′(x)
+if m = 3/2
+3
+2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45
+2 z [g′(x)]2 g′′′(x) + 15zg′(x) [g′′(x)]2
+−15z3g′′(x)g′′′(x) − 60zrg′(x)g′′(x)
+if m = 2
+3
+2z5g(5)(x) − 15z3g(4)(x)g′(x) + 45
+2 z [g′(x)]2 g′′′(x)
++15zg′(x) [g′′(x)]2 − 15z3g′′(x)g′′′(x)
+otherwise
+,
+C6(x, z) = ℓ6
+720
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3
+− 45
+2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x)
+− 45
+2 z4 [g′′′(x)]2 − 90z3rg′(x)g(4)(x) + 540rzg′(x) [g′′(x)]2
++180rzg′(x)g′′(x) + 270rzg′′′(x) [g′(x)]2 − 45zrg′′(x)g′′′(x)
++90rz3g′′′(x)
+if m = 3/2
+2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3
+− 45
+2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x)
+− 45
+2 z4 [g′′′(x)]2 − 180z2rg′(x)g′′′(x) − 180z2r [g′′(x)]2
++180rg′′(x) [g′(x)]2
+if m = 2
+2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3
+− 45
+2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x)
+− 45
+2 z4 [g′′′(x)]2 − 360zrg′(x)g′′(x)
+if m = 5/2
+2z6g(6)(x) − 30z4g′(x)g(5)(x) + 90z2 [g′(x)]2 g(4)(x) − 30g′′′(x) [g′(x)]3
+− 45
+2 [g′(x)g′′(x)]2 − 30z4g′′(x)g(4)(x) + 135z2g′(x)g′′(x)g′′′(x)
+− 45
+2 z4 [g′′′(x)]2
+otherwise
+,
+and integral form of the remainder
+C7(xd
+i , Zi, σd) =
+� σd
+0
+∂7
+∂σ7 R(xd
+i , Zi, σ)|σ=ϵ
+(σd − ϵ)6
+6!
+dϵ,
+with ϵ between 0 and σd and the derivatives of R1 and R2 are given in Appendix C.2.
+65
+
+C.2
+Taylor Expansions of the Log-acceptance Ratio
+C.2.1
+R1
+Recall that R1(x, z, σ) = −g
+�
+x − σ2
+2 g′ �
+proxσ2mr/2
+g
+(x)
+�
++ σz
+�
++ g(x). We compute the derivatives
+of R1 w.r.t. σ evaluated at 0:
+R1(x, z, 0) = 0,
+∂R1
+∂σ (x, z, σ)|σ=0 = −g′(x)z,
+∂2R1
+∂σ2 (x, z, σ)|σ=0 = −z2g′′(x) + [g′(x)]2 ,
+∂3R1
+∂σ3 (x, z, σ)|σ=0 = −z3g′′′(x) + 3g′(x)g′′(x)
+�
+z + ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�
+,
+∂4R1
+∂σ4 (x, z, σ)|σ=0 = −z4g(4)(x) + 6z2g′′′(x)g′(x) − 3g′′(x) [g′(x)]2
++ 12z [g′′(x)]2 ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
++ 6g′(x)
+×
+�
+g′′(x) ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0 +
+� ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�2
+g′′′(x)
+�
+.
+In addition, for m > 1/2 we will also use
+∂5R1
+∂σ5 (x, z, σ)|σ=0 = −z5g(5)(x) + 10z3g(4)(x)g′(x) − 15zg′′′(x) [g′(x)]2
++ 30z [g′′(x)]2 ∂2
+∂2σ proxσ2mr/2
+g
+(x)|σ=0
++ 10g′(x)g′′(x) ∂3
+∂3σ proxσ2mr/2
+g
+(x)|σ=0,
+∂6R1
+∂σ6 (x, z, σ)|σ=0 = −z6g(6)(x) + 15z4g(5)(x)g′(x) − 45z2g(4)(x) [g′(x)]2 + 15g′′′(x) [g′(x)]3
+− 90g′(x) [g′′(x)]2 ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+− 60zg′′(x) ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
++ 90g′′(x)g′′′(x) ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
++ 15g′(x)
+×
+�
+g′′(x) ∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0 + 3
+� ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�2
+g′′′(x)
+�
+.
+66
+
+C.2.2
+R2
+Recall that
+R2(x, z, σ) = 1
+2z2
+− 1
+2
+�
+z − σ
+2 g′
+�
+proxσ2mr/2
+g
+�
+x + σz − σ2
+2 g′ �
+proxσ2mr/2
+g
+(x)
+���
+− σ
+2 g′ �
+proxσ2mr/2
+g
+(x)
+��2
+.
+67
+
+We compute the derivatives of R2 w.r.t. σ evaluated at 0:
+R2(x, z, 0) = 0,
+∂R2
+∂σ (x, z, σ)|σ=0 = zg′(x),
+∂2R2
+∂σ2 (x, z, σ)|σ=0 = − [g′(x)]2 + zg′′(x)
+�
+z + 2 ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�
+,
+∂3R2
+∂σ3 (x, z, σ)|σ=0 = −3g′(x)g′′(x)
+�
+z + 2 ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�
++ 3
+2z
+��
+z + ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�2
+g′′′(x)
++
+�
+−g′(x) + 2z
+∂2
+∂σ∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
+g′′(x)
++
+� ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�2
+g′′′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0g′′(x)
+�
+,
+∂4R2
+∂σ4 (x, z, σ)|σ=0 = −3 [g′′(x)]2
+�
+z + 2 ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�2
+− 6g′(x)
+��
+z + ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�2
+g′′′(x)
++
+�
+−g′(x) + 2z
+∂2
+∂σ∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
+g′′(x)
++
+� ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�2
+g′′′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0g′′(x)
+�
++ 2zg(4)(x)
+�
+z + ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�3
++ 6zg′′′(x)
+�
+z + ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�
+×
+�
+−g′(x) + 2z
+∂2
+∂σ∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 2zg′′(x)
+�
+−3g′′(x) ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0 − 3g′(x) ∂2
+∂σ∂x proxσ2mr/2
+g
+(x)|σ=0
++3z2
+∂3
+∂σ∂x2 proxσ2mr/2
+g
+(x)|σ=0 + 3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0
++ ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 2zg(4)(x)
+� ∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0
+�3
++ 2zg′′(x) ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
++ 6z ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0g′′′(x).
+68
+
+We then proceed to get the derivatives needed for m > 1/2:
+∂5R2
+∂σ5 (x, z, σ)|σ=0 = −15zg′′(x)
+�
+z2g′′′(x) +
+�
+−g′(x) + 2 ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
+g′′(x)
+�
+− 5g′(x)
+�
+2z3g(4)(x) + 6zg′′′(x)
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
++2g′′(x)
+�
+3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
++2g′′(x) ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 5
+2g(5)(x)z5 + 5
+2zg′′(x) ∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0
++ 15g(4)(x)z3
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 15
+2 zg′′′(x)
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�2
++ 15
+2 zg′′′(x)
+� ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�2
++ 10z2g′′′(x)
+�
+3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 5
+2zg′′(x)
+� ∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0 − 6g′′(x) ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+−6g′(x)
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0 + 6z2
+∂4
+∂σ2∂x2 proxσ2mr/2
+g
+(x)|σ=0
++4z
+∂4
+∂σ3∂x proxσ2mr/2
+g
+(x)|σ=0
+�
+and
+∂6R2
+∂σ6 (x, z, σ)|σ=0 = −45
+2
+�
+z2g′′′(x) +
+�
+−g′(x) + 2 ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
+g′′(x)
+�2
+− 15zg′′(x)
+�
+2z3g(4)(x) + 6zg′′′(x)
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 2g′′(x)
+�
+3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
++2g′′(x) ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 6g′(x)A(5) − zA(6),
+69
+
+with
+A(5) = −5
+2g(5)(x)z4 − 5
+2g′′(x) ∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0
+− 15g(4)(x)z2
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
+− 15
+2 g′′′(x)
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�2
+− 15
+2 g′′′(x)
+� ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�2
+− 10zg′′′(x)
+�
+3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0 + ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
+�
+− 5
+2g′′(x)
+� ∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0 − 6g′′(x) ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+−6g′(x)
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0
++6z2
+∂4
+∂σ2∂x2 proxσ2mr/2
+g
+(x)|σ=0 + 4z
+∂4
+∂σ3∂x proxσ2mr/2
+g
+(x)|σ=0
+�
+,
+A(6) = −3
+�
+10g′′′(x) ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0
++g′′(x) ∂5
+∂σ5 proxσ2mr/2
+g
+(x)|σ=0 + g(6)(x)z5
++ 10g(5)(x)z3
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
++ 15zg(4)(x)
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�2
++ 10g(4)(x)z2
+� ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0 + 3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0
+�
++ 10g′′′(x)
+�
+−g′(x) + ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0
+�
+×
+� ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0 + 3z
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0
+�
++ 5g′′′(x)z
+�
+−6g′′(x) ∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0 − 6g′(x)
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0
++6z2
+∂4
+∂σ2∂x2 proxσ2mr/2
+g
+(x)|σ=0 + 3z
+∂4
+∂σ3∂x proxσ2mr/2
+g
+(x)|σ=0
++ ∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0
+�
++ g′′(x)
+�
+−10g′′(x) ∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0 − 10g′(x)
+∂4
+∂σ3∂x proxσ2mr/2
+g
+(x)|σ=0
++ 5z
+∂5
+∂σ4∂x proxσ2mr/2
+g
+(x)|σ=0 + 10z2
+∂5
+∂σ3∂x2 proxσ2mr/2
+g
+(x)|σ=0
++ 10z3
+∂5
+∂σ2∂x3 proxσ2mr/2
+g
+(x)|σ=0
+−30g′(x)z
+∂4
+∂σ2∂x2 proxσ2mr/2
+g
+(x)|σ=0 + ∂5
+∂σ5 proxσ2mr/2
+g
+(x)|σ=0
+��
+.
+70
+
+C.3
+Derivatives of the Proximity Map for Differentiable Targets
+Recall that, in the differentiable case, proxσ2mr/2
+g
+(x) = − σ2mr
+2
+g′(proxσ2mr/2
+g
+(x)) + x then
+∂
+∂σ proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+− r
+2g′(x)
+if m = 1/2
+0
+if m > 1/2
+∞
+otherwise
+∂2
+∂σ2 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+r2
+2 g′(x)g′′(x)
+if m = 1/2
+−rg′(x)
+if m = 1
+0
+if m > 1
+∞
+otherwise
+∂3
+∂σ3 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− 3r3
+8 g′′′(x) [g′(x)]2 − 3r3
+4 g′(x) [g′′(x)]2
+if m = 1/2
+−3rg′(x)
+if m = 3/2
+0
+if m = 1, m > 3/2
+∞
+otherwise
+∂4
+∂σ4 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+6r2g′(x)g′′(x)
+if m = 1
+−12rg′(x)
+if m = 2
+0
+if m = 3/2, m > 2
+∞
+otherwise
+∂5
+∂σ5 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+−60rg′(x)
+if m = 5/2
+0
+if m = 1, m = 3/2, m = 2, m > 5/2
+∞
+otherwise
+and
+∂
+∂x proxσ2mr/2
+g
+(x)|σ=0 = 1,
+∂(k)
+∂x(k) proxσ2mr/2
+g
+(x)|σ=0 = 0,
+71
+
+for all integers k > 1. For the mixed derivatives we have
+∂2
+∂σ∂x proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+− r
+2g′′(x)
+if m = 1/2
+0
+if m > 1/2
+∞
+otherwise
+∂3
+∂σ2∂x proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+r2
+2 [g′′(x)]2 + r2
+2 g′(x)g′′′(x)
+if m = 1/2
+−rg′′(x)
+if m = 1
+0
+if m > 1/2
+∞
+otherwise
+∂4
+∂σ3∂x proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+−3rg′′(x)
+if m = 3/2
+0
+if m = 1, m > 3/2
+∞
+otherwise
+∂5
+∂σ4∂x proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+6r2 �
+g′(x)g′′′(x) + [g′′(x)]2�
+if m = 1
+−12rg′′(x)
+if m = 2
+0
+if m = 3/2, m > 2
+∞
+otherwise
+72
+
+and
+∂3
+∂σ∂x2 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+− r
+2g′′′(x)
+if m = 1/2
+0
+if m > 1/2
+∞
+otherwise
+∂4
+∂σ∂x3 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+0
+if m > 1/2
+∞
+otherwise
+∂4
+∂σ2∂x2 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+−rg′′′(x)
+if m = 1
+0
+if m > 1/2
+∞
+otherwise
+∂5
+∂σ3∂x2 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+−3rg′′′(x)
+if m = 3/2
+0
+if m = 1, m > 3/2
+∞
+otherwise
+∂5
+∂σ2∂x3 proxσ2mr/2
+g
+(x)|σ=0 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+< ∞
+if m = 1/2
+−rg(4)(x)
+if m = 1
+0
+if m > 1
+∞
+otherwise
+D
+Moments and Integrals for the Laplace Distribution
+D.1
+Moments of Acceptance Ratio for the Laplace Distribution
+The indicator functions in the definition of φd identify four different regions:
+R1 :=
+�
+(x, z) : |x| ≤ σ2mr/2 ∧
+����
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz
+���� ≤ σ2mr/2
+�
+,
+R2 :=
+�
+(x, z) : |x| > σ2mr/2 ∧
+����x − σ2
+2 sgn(x) + σz
+���� ≤ σ2mr/2
+�
+,
+R3 :=
+�
+(x, z) : |x| ≤ σ2mr/2 ∧
+����
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz
+���� > σ2mr/2
+�
+,
+R4 :=
+�
+(x, z) : |x| > σ2mr/2 ∧
+����x − σ2
+2 sgn(x) + σz
+���� > σ2mr/2
+�
+,
+73
+
+with corresponding acceptance ratios
+φ1
+d(x, z) = |x| −
+����
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz
+���� + z2
+2
+−
+1
+2σ2
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+x −
+�
+1 −
+1
+σ2(m−1)r
+�
+σz
+�2
+φ2
+d(x, z) = |x| −
+����x − σ2
+2 sgn(x) + σz
+���� + z2
+2
+−
+1
+2σ2
+�
+1
+σ2(m−1)rx +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 sgn(x) − σz
+��2
+φ3
+d(x, z) = |x| −
+����
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz
+���� + z2
+2
+−
+1
+2σ2
+�
+1
+σ2(m−1)rx − σz + σ2
+2 sgn
+��
+1 −
+1
+σ2(m−1)r
+�
+x + σz
+��2
+φ4
+d(x, z) = |x| −
+����x − σ2
+2 sgn(x) + σz
+���� + z2
+2
+−
+1
+2σ2
+�σ2
+2 sgn(x) − σz + σ2
+2 sgn
+�
+x − σ2
+2 sgn(x) + σz
+��2
+.
+Let us denote
+A1 :=
+�
+x : 0 ≤ x ≤ σ2mr
+2
+�
+,
+A2 :=
+�
+x : −σ2mr
+2
+≤ x ≤ 0
+�
+,
+A3 :=
+�
+x : x > σ2mr
+2
+�
+,
+A4 :=
+�
+x : x < −σ2mr
+2
+�
+,
+and
+B1 :=
+�
+z : 0 ≤
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz ≤ σ2mr
+2
+�
+,
+B2 :=
+�
+z : −σ2mr
+2
+≤
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz ≤ 0
+�
+,
+B3 :=
+�
+z :
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz > σ2mr
+2
+�
+,
+B4 :=
+�
+z :
+�
+1 −
+1
+σ2(m−1)r
+�
+x + σz < −σ2mr
+2
+�
+,
+74
+
+and
+C1 :=
+�
+(x, z) : 0 ≤ x − σ2
+2 sgn(x) + σz ≤ σ2mr
+2
+�
+,
+C2 :=
+�
+(x, z) : −σ2mr
+2
+≤ x − σ2
+2 sgn(x) + σz ≤ 0
+�
+C3 :=
+�
+(x, z) : x − σ2
+2 sgn(x) + σz > σ2mr
+2
+�
+,
+C4 :=
+�
+(x, z) : x − σ2
+2 sgn(x) + σz < −σ2mr
+2
+�
+,
+so that
+R1 = (A1 ∪ A2) ∩ (B1 ∪ B2),
+R2 = (A3 ∪ A4) ∩ (C1 ∪ C2),
+R3 = (A1 ∪ A2) ∩ (B3 ∪ B4),
+R4 = (A3 ∪ A4) ∩ (C3 ∪ C4).
+Proposition 16. Take X a Laplace random variable and Z a standard normal random variable
+independent of X, then if σ2 = ℓ2d−2/3, we have
+lim
+d→+∞ dE [φd(X, Z)] = −
+ℓ3
+3
+√
+2π.
+Proof. Taking expectations of φi
+d1Ri for i = 1, . . . , 4 and exploiting the symmetry of the laws of X
+75
+
+and Z, we can write
+E
+�
+φ1
+d(X, Z)1R1(X, Z)
+�
+= 2E
+��
+1
+σ2(m−1)rX − σZ
+�
+1A1(X)1B1(X, Z)
+�
++ 2E
+��
+2X −
+1
+σ2(m−1)rX + σZ
+�
+1A1(X)1B2(X, Z)
+�
++ 2E
+��
+Z2
+2 −
+1
+2σ2
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+X −
+�
+1 −
+1
+σ2(m−1)r
+�
+σZ
+�2�
+×1A1(X)1B1∪B2(X, Z)] ,
+E
+�
+φ2
+d(X, Z)1R2(X, Z)
+�
+= 2E
+��σ2
+2 − σZ
+�
+1A3(X)1C1(X, Z)
+�
++ 2E
+��
+2X − σ2
+2 + σZ
+�
+1A3(X)1C2(X, Z)
+�
++ 2E
+��
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σZ
+��2�
+×1A3(X)1C1∪C2(X, Z)] ,
+E
+�
+φ3
+d(X, Z)1R3(X, Z)
+�
+= 2E
+��
+1
+σ2(m−1)rX − σZ
+�
+1A1(X)1B3(X, Z)
+�
++ 2E
+��
+2X −
+1
+σ2(m−1)rX + σZ
+�
+1A1(X)1B4(X, Z)
+�
++ 2E
+��
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ + σ2
+2
+�2�
+1A1(X)1B3(X, Z)
+�
++ 2E
+��
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ − σ2
+2
+�2�
+1A1(X)1B4(X, Z)
+�
+,
+E
+�
+φ4
+d(X, Z)1R4(X, Z)
+�
+= 2E
+��
+2X − σ2
+2 + σZ
+�
+1A3(X)1C4(X, Z)
+�
+.
+Using the integrals in Appendix D.4 and Lebesgue’s dominated convergence theorem, we find that
+76
+
+for α = β = 1/3 and r ≥ 0
+lim
+d→+∞ dE
+�
+φ1
+d(X, Z)
+�
+= 0
+lim
+d→+∞ dE
+�
+φ2
+d(X, Z)
+�
+= −2 ℓ3r
+4
+√
+2π
+� 0
+−∞
+e−z2/2zdz =
+ℓ3r
+2
+√
+2π
+lim
+d→+∞ dE
+�
+φ3
+d(X, Z)
+�
+= 3ℓ3r
+8
+√
+2π
+� 0
+−∞
+e−z2/2zdz −
+ℓ3r
+8
+√
+2π
+� +∞
+0
+e−z2/2zdz = − ℓ3r
+2
+√
+2π
+lim
+d→+∞ dE
+�
+φ4
+d(X, Z)
+�
+=
+ℓ3
+6
+√
+2π
+� 0
+−∞
+e−z2/2z3dz = −
+ℓ3
+3
+√
+2π,
+which gives
+lim
+d→+∞ dE [φd(X, Z)] =
+lim
+d→+∞ d
+�
+E
+�
+φ1
+d(X, Z)
+�
++ E
+�
+φ2
+d(X, Z)
+�
++ E
+�
+φ3
+d(X, Z)
+�
++ E
+�
+φ4
+d(X, Z)
+��
+= −
+ℓ3
+3
+√
+2π.
+For α = 1/3, β = m/3 for m > 1 and r ≥ 0 we have
+lim
+d→+∞ dE
+�
+φ1
+d(X, Z)
+�
+= 0
+lim
+d→+∞ dE
+�
+φ2
+d(X, Z)
+�
+= 0
+lim
+d→+∞ dE
+�
+φ3
+d(X, Z)
+�
+= 0
+lim
+d→+∞ dE
+�
+φ4
+d(X, Z)
+�
+=
+ℓ3
+6
+√
+2π
+� 0
+−∞
+e−z2/2z3dz = −
+ℓ3
+3
+√
+2π,
+which gives
+lim
+d→+∞ dE [φd(X, Z)] =
+lim
+d→+∞ d
+�
+E
+�
+φ1
+d(X, Z)
+�
++ E
+�
+φ2
+d(X, Z)
+�
++ E
+�
+φ3
+d(X, Z)
+�
++ E
+�
+φ4
+d(X, Z)
+��
+= −
+ℓ3
+3
+√
+2π.
+Proposition 17. Take X a Laplace random variable and Z a standard normal random variable
+independent of X, then if σ2 = ℓ2d−2/3
+lim
+d→+∞ d Var (φd(X, Z)) =
+2ℓ3
+3
+√
+2π.
+Proof. As a consequence of the previous Proposition we have
+lim
+d→+∞ dE [φd(X, Z)]2 = 0.
+77
+
+Then, because Rj ∩ Ri = ∅ for all j ̸= i, we have that
+E
+�
+φ(X, Z)2�
+= E
+�
+φ1
+d(X, Z)2R1(X, Z)
+�
++ E
+�
+φ2
+d(X, Z)2R2(X, Z)
+�
++ E
+�
+φ3
+d(X, Z)2R3(X, Z)
+�
++ E
+�
+φ4
+d(X, Z)2R4(X, Z)
+�
+,
+and, exploiting again the symmetry of the laws of X and Z, we have
+E
+�
+φ1
+d(X, Z)2R1(X, Z)
+�
+= 2E
+�
+�
+�
+1
+σ2(m−1)rX − σZ + Z2
+2 −
+1
+2σ2
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+X −
+�
+1 −
+1
+σ2(m−1)r
+�
+σZ
+�2�2
+×1A1(X)1B1(X, Z)] ,
++ 2E
+�
+�
+�
+2X −
+1
+σ2(m−1)rX − σZ + Z2
+2 −
+1
+2σ2
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+X −
+�
+1 −
+1
+σ2(m−1)r
+�
+σZ
+�2�2
+×1A1(X)1B2(X, Z)] ,
+E
+�
+φ2
+d(X, Z)21R2(X, Z)
+�
+= 2E
+�
+�
+�
+σ2
+2 − σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σZ
+��2�2
+1A3(X)1C1(X, Z)
+�
+�
++ 2E
+�
+�
+�
+2X − σ2
+2 + σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σZ
+��2�2
+×1A1(X)1C2(X, Z)] ,
+E
+�
+φ3
+d(X, Z)21R3(X, Z)
+�
+= 2E
+�
+�
+�
+1
+σ2(m−1)rX − σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ + σ2
+2
+�2�2
+1A1(X)1B3(X, Z)
+�
+�
++ 2E
+�
+�
+�
+2X −
+1
+σ2(m−1)rX + σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ − σ2
+2
+�2�2
+1A1(X)1B4(X, Z)
+�
+� ,
+E
+�
+φ4
+d(X, Z)21R4(X, Z)
+�
+= 2E
+��
+2X − σ2
+2 + σZ
+�2
+1A3(X)1C4(X, Z)
+�
+.
+Proceeding as for Proposition 16, using the integrals in Appendix D.4 and Lebesgue’s dominated
+convergence theorem we can then show that for α = 1/3, β = m/3 for m ≥ 1 and r ≥ 0
+lim
+d→+∞ d Var (φd(X, Z)) =
+2ℓ3
+3
+√
+2π.
+78
+
+Proposition 18. Take X a Laplace random variable and Z a standard normal random variable
+independent of X, then if σ2 = ℓ2d−2/3 we have
+lim
+d→+∞ dE
+�
+φd(X, Z)3�
+= 0.
+Proof. Following the same structure of the previous propositions we have that
+E
+�
+φ(X, Z)3�
+= E
+�
+φ1
+d(X, Z)3R1(X, Z)
+�
++ E
+�
+φ3
+d(X, Z)2R2(X, Z)
+�
++ E
+�
+φ3
+d(X, Z)3R3(X, Z)
+�
++ E
+�
+φ4
+d(X, Z)3R4(X, Z)
+�
+,
+exploiting again the symmetry of the laws of X and Z, using the integrals in Appendix D.4, the
+dominated convergence theorem we can then show that
+lim
+d→+∞ dE
+�
+φd(X, Z)3�
+= 0.
+D.2
+Bound on Second Moment of Acceptance Ratio for the Laplace Dis-
+tribution
+Lemma 3. Let Z be a standard normal random variable and σ = ℓ/dα for α = 1/3. Then, there
+exists a constant C > 0 such that for all a ∈ R and d ∈ N:
+E
+�
+φd(a, Z)2�
+≤ C
+d2α .
+Proof. We consider the case a ≥ 0 and r ≥ σ2(m−1) only, all the other cases follow from identical
+arguments. As in the derivation of the moments of φd in Appendix D.1, we distinguish four regions.
+We recall that σ = ℓ/dα for α = 1/3 and thus σp+1 ≤ σp for all p ∈ N. Take r ≥ σ−2(m−1), for R1,
+79
+
+we have, using H¨older’s inequality multiple times,
+E
+�
+φ1
+d(a, Z)21R1(a, Z)
+�
+= E
+�
+φ1
+d(a, Z)21B1∪B2(a, Z)
+�
+≤ Cσ2
+� σ2m−1r/2−(1−1/σ2(m−1)r)a
+−(1−1/σ2(m−1)r)a/σ
+e−z2/2
+√
+2π
+��
+1
+σ2m−1ra − z
+�2
++
+�
+z2
+2σ −
+1
+2σ3
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+a −
+�
+1 −
+1
+σ2(m−1)r
+�
+σz
+�2�2�
+� dz
++ Cσ2
+� −(1−1/σ2(m−1)r)a/σ
+−σ2m−1r/2−(1−1/σ2(m−1)r)a
+e−z2/2
+√
+2π
+��2a
+σ −
+1
+σ2m−1ra + z
+�2
++
+�
+z2
+2σ −
+1
+2σ3
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+a −
+�
+1 −
+1
+σ2(m−1)r
+�
+σz
+�2�2�
+� dz
+≤ Cσ2
+� +∞
+−∞
+e−z2/2
+√
+2π
+�
+4
+� a
+σ
+�2
++ 2
+�
+1
+σ2m−1ra
+�2
++ 2z2
+�
+dz
++ Cσ2
+� σ2m−1r/2−(1−1/σ2(m−1)r)a
+−σ2m−1r/2−(1−1/σ2(m−1)r)a
+e−z2/2
+√
+2π
+×
+�
+z2
+2σ −
+1
+2σ3
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�
+a −
+�
+1 −
+1
+σ2(m−1)r
+�
+σz
+�2�2
+≤ Cσ2 + Cσ2
+� σ2m−1r/2−(1−1/σ2(m−1)r)a
+−σ2m−1r/2−(1−1/σ2(m−1)r)a
+e−z2/2
+√
+2π
+×
+�
+z4
+4σ2 +
+1
+4σ6
+��
+2
+σ2(m−1)r −
+1
+σ4(m−1)r2
+�4
+a4 +
+�
+1 −
+1
+σ2(m−1)r
+�4
+σ4z4
+��
+dz
+≤ Cσ2,
+where we used the fact that the moments of Z are bounded and a ≤ σ2mr/2 for the first term,
+and the fact that z ≤ σ2m−1r/2 for the second one.
+Proceeding as above, for R3, a > 0 and
+80
+
+r ≥ σ−2(m−1), we have
+E
+�
+φ3
+d(a, Z)21R3(a, Z)
+�
+= E
+�
+φ3
+d(a, Z)21B3∪B4(a, Z)
+�
+≤ Cσ2
+� +∞
+σ2m−1r/2−(1−1/σ2(m−1)r)a/σ
+e−z2/2
+√
+2π
+��
+1
+σ2m−1ra − z
+�2
++
+�
+z2
+2σ −
+1
+2σ3
+�
+1
+σ2(m−1)ra − σz + σ2
+2
+�2�2�
+� dz
++ Cσ2
+� −σ2m−1r/2−(1−1/σ2(m−1)r)a/σ
+−∞
+e−z2/2
+√
+2π
+��2a
+σ −
+1
+σ2m−1ra + z
+�2
++
+�
+z2
+2σ −
+1
+2σ3
+�
+1
+σ2(m−1)ra − σz + σ2
+2
+�2�2�
+� dz
+≤ Cσ2 + Cσ2
+� +∞
+−∞
+e−z2/2
+√
+2π
+� 1
+2σ3
+�
+a2
+σ4(m−1)r2 + σ4
+4 − σ3z +
+a
+σ2m−4r −
+2az
+σ2m−3r
+��2
+dz
+≤ Cσ2,
+where we used again the boundedness of the moments of Z, the fact that a ≤ σ2mr/2 and that
+σp+1 ≤ σp. For R2 and a > 0, we have
+E
+�
+φ2
+d(a, Z)21R2(a, Z)
+�
+= E
+�
+φ2
+d(a, Z)21C1∪C2(a, Z)
+�
+≤ Cσ2
+� σ/2+σ2m−1r/2−a/σ
+σ/2−a/σ
+e−z2/2
+√
+2π
+��σ2
+2 − σz
+�2
++
+�
+z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)ra +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σz
+��2�2�
+� dz
++ Cσ2
+� σ/2−a/σ
+σ/2−σ2m−1r/2−a/σ
+e−z2/2
+√
+2π
+��
+2a − σ2
+2 + σz
+�2
++
+�
+z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)ra +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σz
+��2�2�
+� dz.
+The first integral is bounded using the moments of Z, while for the third one let us denote
+81
+
+χ(a, σ, z) := a − σ2/2 + σz, then
+� σ/2+σ2m−1r/2−a/σ
+σ/2−σ2m−1r/2−a/σ
+e−z2/2
+√
+2π
+�
+z2
+2σ −
+1
+2σ3
+�
+1
+σ2(m−1)ra +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σz
+��2�2
+dz
+=
+� σ/2+σ2m−1r/2−a/σ
+σ/2−σ2m−1r/2−a/σ
+e−z2/2
+√
+2π
+�
+z2
+2σ −
+1
+2σ3
+�χ(a, σ, z)
+σ2(m−1)r + σ2
+2 − σz
+�2�2
+dz
+≤ C
+� σ/2+σ2m−1r/2−a/σ
+σ/2−σ2m−1r/2−a/σ
+e−z2/2
+√
+2π
+�
+z2
+2σ −
+1
+2σ3
+�σ2
+2 − σz
+�2�2
+dz
++ C
+� σ/2+σ2m−1r/2−a/σ
+σ/2−σ2m−1r/2−a/σ
+e−z2/2
+√
+2π
+�χ(a, σ, z)2
+2σ4m−1r2
+�2
++
+�χ(a, σ, z)
+rσ2m−1
+�σ2
+2 − σz
+��2
+dz;
+recalling that in R2 we have |χ(a, σ, z)| ≤ σ2mr/2, we obtain that this term is also bounded by
+Cσ2. For R4 and a > 0, we have
+E
+�
+φ4
+d(a, Z)21R4(a, Z)
+�
+= E
+�
+φ4
+d(a, Z)21C4(a, Z)
+�
+=
+� σ/2−σ2m−1r/2−a/σ
+−∞
+e−z2/2
+√
+2π
+�
+2a − σ2
+2 + σz
+�2
+dz
+= σ2
+� σ/2−σ2m−1r/2−a/σ
+−∞
+e−z2/2
+√
+2π
+�2a
+σ − σ
+2 + z
+�2
+dz
+Collecting all the terms together, we obtain
+E
+�
+φd(a, Z)2�
+=
+4
+�
+i=1
+E
+�
+φi
+d(a, Z)21Ti(a, Z)
+�
+≤ Cσ2 + Cσ2
+� σ/2−a/σ
+−∞
+e−z2/2
+√
+2π
+�2a
+σ − σ
+2 + z
+�2
+dz.
+Recall that σ = ℓd−1/3. To bound the last integral we use H¨older’s inequality
+� σ/2−a/σ
+−∞
+e−z2/2
+√
+2π
+�2a
+σ − σ
+2 + z
+�2
+dz ≤ C
+� σ/2−a/σ
+−∞
+e−z2/2
+√
+2π
+��σ
+2 + z
+�2
++ 4
+�σ
+2 − a
+σ
+�2�
+dz
+≤ C
+� σ/2−a/σ
+−∞
+e−z2/2
+√
+2π
+��ℓ2
+4 + z2
+�
++ 4
+�σ
+2 − a
+σ
+�2�
+dz.
+The first term is bounded since the moments of Z are bounded. For the second term we use an
+estimate of the Gaussian cumulative distribution function. Let κ(ℓ, d, a) := ℓd−1/3/2 − ad1/3/ℓ.
+When z < κ(ℓ, d, a) < 0, we have 1 < z/κ(ℓ, d, a) and therefore
+(2π)−1/2κ(ℓ, d, a)2
+� κ(ℓ,d,a)
+−∞
+e−z2/2dz ≤ (2π)−1/2κ(ℓ, d, a)
+� κ(ℓ,d,a)
+−∞
+ze−z2/2dz,
+= (2π)−1/2κ(ℓ, d, a) exp(−κ(ℓ, d, a)2/2).
+82
+
+However y �→ ye−y2/2 is bounded over R, therefore (a, d) �−→ (2π)−1/2κ(ℓ, d, a) exp(−κ(ℓ, d, a)2/2)
+is bounded over R∗
++ × N.
+If κ(ℓ, d, a) ≥ 0, then we still have κ(ℓ, d, a) < ℓ and thus have the
+inequality
+(2π)−1/2κ(ℓ, d, a)2
+� κ(ℓ,d,a)
+−∞
+e−z2/2dz ≤ (2π)−1/2ℓ2
+� +∞
+−∞
+e−z2/2dz = ℓ2.
+The result then follows since σ = ℓd−1/3.
+D.3
+Additional Integrals for the Laplace Distribution
+We collect here two auxiliary Lemmata which are used in the proof of Proposition 3.
+Lemma 4. Take X a Laplace random variable and Z a standard normal random variable indepen-
+dent of X. Let ˜X := X −
+1
+σ2(m−1)rX1|X|≤σ2mr/2 − σ2
+2 sgn(X)1|X|>σ2mr/2 + σZ, then, for σ = ℓd−α
+with α = 1/3,
+E
+�
+1{sgn(X)̸=sgn( ˜
+X)}
+�
+→ 0
+if d → ∞.
+Proof. Using the same strategy of Appendix D.1 and the symmetry of the laws of X, Z, we find
+that
+E
+�
+1{sgn(X)̸=sgn( ˜
+X)}
+�
+= 2E [1A1(X)1B2(X, Z)] + 2E [1A3(X)1C2(X, Z)]
++ 2E [1A1(X)1B4(X, Z)] + 2E [1A3(X)1C4(X, Z)] .
+Using the same strategy used to obtain the moments of φd in Appendix D.1, we find that
+E [1A1(X)1B2(X, Z)] = o(1),
+in addition
+E [1A3(X)1C2(X, Z)] =
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2
+� σ2/2−σz+σ2mr/2
+σ2/2−σz
+e−xdx dz + o(1)
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2
+�σ2r
+2 δm1 + ...
+�
+dz + o(1),
+where δm1 is a Dirac’s delta, and
+E [1A1(X)1B4(X, Z)] + E [1A3(X)1C4(X, Z)]
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2
+� σ2/2−σz−σ2mr/2
+0
+e−xdx dz + o(1)
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2 [−σz + ...] dz + o(1).
+83
+
+Since σ = ℓd−1/3 and the remainder terms of the Taylor expansions are bounded, Lebesque’s
+dominated convergence theorem gives
+E [1A3(X)1C2(X, Z)] → 0,
+E [1A1(X)1B4(X, Z)] + E [1A3(X)1C4(X, Z)] → 0
+as d → ∞.
+Lemma 5. Take X a Laplace random variable and Z a standard normal random variable indepen-
+dent of X. Then,
+dαE [|Z| |φd (X, Z)|] → 0
+for α = 1/3.
+Proof. Using Cauchy-Schwarz’s inequality we have that
+E [|Z| |φd (X, Z)|] ≤ E
+�
+Z2�1/2 E
+�
+φd(X, Z)2�1/2 ;
+the first expectation is equal to one, and the second one converges to zero at rate d1/2 by Proposi-
+tion 17. The result follows straightforwardly.
+D.4
+Integrals for Moment Computations
+We distinguish the case m = 1/2 and m ≥ 1 since the integration bounds significantly differ in
+these two cases. For values between 1/2 and 1 the integrals are not finite. The expectations below
+are obtained by integrating w.r.t. x and using a Taylor expansion about σ = 0 to obtain the leading
+order terms. Using the Lagrange form of the remainder for the Taylor expansions, we find that the
+remainder terms are all of the form σ1/α+1f(γ(σ, z))/(1/α + 1)! where γ(σ, z) is a point between
+the limits of integration w.r.t. x and f : x �→ p(x)e−x, where p is a polynomial. Therefore, using
+the boundedness of the remainder and Lebesgue’s dominated convergence theorem, the integrals
+w.r.t. z of the remainder terms all converge to 0.
+D.4.1
+First Moment
+For simplicity, we only consider the case for r ≥ σ−2(m−1), the other case follows analogously.
+Region R1
+Let us consider φ1
+d first. We have
+A1 ∩ B1 =
+�
+�
+�
+�
+�
+�
+�
+0 ≤ x ≤ σ2mr
+2
+if 0 ≤ z ≤ σ
+2
+0 ≤ x ≤
+�
+σ2mr
+2
+− σz
+� �
+1 −
+1
+σ2(m−1)r
+�−1
+if σ
+2 ≤ z ≤ σ2m−1r
+2
+−σz
+�
+1 −
+1
+σ2(m−1)r
+�−1 ≤ x ≤ σ2mr
+2
+if σ
+2 − σ2m−1r
+2
+≤ z ≤ 0
+,
+A1 ∩ B2 =
+�
+�
+�
+�
+�
+�
+�
+0 ≤ x ≤ σ2mr
+2
+if − σ2m−1r
+2
+≤ z ≤ σ
+2 − σ2m−1r
+2
+0 ≤ x ≤ −σz
+�
+1 −
+1
+σ2(m−1)r
+�−1
+if σ
+2 − σ2m−1r
+2
+≤ z ≤ 0
+−
+�
+σ2mr
+2
++ σz
+� �
+1 −
+1
+σ2(m−1)r
+�−1 ≤ x ≤ σ2mr
+2
+if σ
+2 − σ2m−1r ≤ z ≤ − σ2m−1r
+2
+,
+84
+
+and
+A1 ∩ (B1 ∪ B2) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0 ≤ x ≤ σ2mr
+2
+if − σ2m−1r
+2
+≤ z ≤ σ
+2
+0 ≤ x ≤
+�
+σ2mr
+2
+− σz
+� �
+1 −
+1
+σ2(m−1)r
+�−1
+if σ
+2 ≤ z ≤ σ2m−1r
+2
+−
+�
+σ2mr
+2
++ σz
+� �
+1 −
+1
+σ2(m−1)r
+�−1 ≤ x ≤ σ2mr
+2
+if σ
+2 − σ2m−1r ≤ z ≤ − σ2m−1r
+2
+.
+The corresponding expectations are
+E
+��
+X
+σ2(m−1)r − σZ
+�
+1A1(X)1B1(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ2m−1r/2
+σ/2
+e−z2/2 �
+z4−2mσ4−2mξ(r) + . . .
+�
+dz
++
+1
+2
+√
+2π
+� 0
+σ/2−rσ2m−1/2
+e−z2/2 �
+z4−2mσ4−2mξ(r) + . . .
+�
+dz
++ o(1),
+E
+��
+2X −
+X
+σ2(m−1)r + σZ
+�
+1A1(X)1B2(X, Z)
+�
+=
+1
+2
+√
+2π
+� 0
+σ/2−σ2m−1r/2
+e−z2/2 �
+z4−2mσ4−2mξ(r) + . . .
+�
+dz
++
+1
+2
+√
+2π
+� −σ2m−1r/2
+σ/2−σ2m−1r
+e−z2/2 �
+z4−2mσ4−2mξ(r) + . . .
+�
+dz
++ o(1),
+where ξ : [0, +∞) → R is a function of r only which might change from one line to the other, and
+E
+��
+Z2
+2 −
+1
+2σ2
+��
+2
+σ2(m−1)r −
+1
+σ4(m���1)r2
+�
+X −
+�
+1 −
+1
+σ2(m−1)r
+�
+σZ
+�2�
+1A1(X)1B1∪B2(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2
+−σ2m−1r/2
+e−z2/2 [+ . . . ] dz
++
+1
+2
+√
+2π
+� σ2m−1r/2
+σ/2
+e−z2/2 [+ . . . ] dz
++
+1
+2
+√
+2π
+� −σ2m−1r/2
+σ/2−σ2m−1r
+e−z2/2 �
+z2σ2ξ(r) + . . .
+�
+dz,
+where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.
+85
+
+Region R2
+For φ2
+d, we have
+A3 ∩ C1 =
+�
+σ2/2 − σz ≤ x ≤ −σz + σ2/2 + σ2mr/2
+if z < σ/2 − σ2m−1r/2
+σ2mr/2 < x ≤ −σz + σ2/2 + σ2mr/2
+if σ/2 − σ2m−1r/2 ≤ z ≤ σ/2
+,
+A3 ∩ C2 =
+�
+σ2/2 − σz − σ2mr/2 ≤ x ≤ σ2/2 − σz
+if z < σ/2 − σ2m−1r
+σ2mr/2 < x ≤ σ2/2 − σz
+if σ/2 − σ2m−1r < z < σ/2 − σ2m−1r/2
+and
+A3 ∩ (C1 ∪ C2) =
+�
+�
+�
+�
+�
+σ2mr/2 ≤ x ≤ σ2mr/2 + σ2/2 − σz
+if σ/2 − σ2m−1r ≤ z ≤ σ/2
+−σ2mr/2 + σ2/2 − σz < x ≤ σ2mr/2 + σ2/2 − σz
+if z < σ/2 − σ2m−1r
+.
+The corresponding expectations are
+E
+��σ2
+2 − σZ
+�
+1A3(X)1C1(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2 �
+−rz
+2 σ2m+1 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ/2
+σ/2−σ2m−1r/2
+e−z2/2 �
+z2σ2 + . . .
+�
+dz,
+E
+��
+2X − σ2
+2 + σZ
+�
+1A3(X)1C2(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2 �
+−rz
+2 σ2m+1 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+σ/2−σ2m−1r
+e−z2/2 �
+−rz
+2 σ2m+1 + . . .
+�
+dz,
+and
+E
+��
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σZ
+��2�
+1A3(X)1C1∪C2(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2 �rz
+2 σ2m+1 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ/2
+σ/2−σ2m−1r
+e−z2/2
+�
+−z3
+2rσ3−2m + . . .
+�
+dz.
+Region R3
+For φ3
+d, we have, in the case r ≥ σ−2(m−1),
+A1 ∩ B3 =
+�
+0 ≤ x ≤ σ2mr
+2
+if z > σ2m−1r
+2
+�
+σ2mr
+2
+− σz
+� �
+1 −
+1
+σ2(m−1)r
+�−1 < x ≤ σ2mr
+2
+if σ
+2 ≤ z ≤ σ2m−1r
+2
+,
+A1 ∩ B4 =
+�
+0 ≤ x ≤ σ2mr
+2
+if z < σ
+2 − σ2m−1r
+0 ≤ x <
+�
+σ2mr
+2
+− σz
+� �
+1 −
+1
+σ2(m−1)r
+�−1
+if σ
+2 − σ2m−1r ≤ z ≤ − σ2m−1r
+2
+.
+86
+
+The corresponding expectations are
+E
+��
+1
+σ2(m−1)rX − σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ + σ2
+2
+�2�
+1A1(X)1B3(X, Z)
+�
+=
+1
+2
+√
+2π
+� +∞
+σ2m−1r/2
+e−z2/2 �
+−rz
+8 σ2m+1 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ2m−1r/2
+σ/2
+e−z2/2 �
+z3−2mσ3−2mξ(r) + . . .
+�
+dz,
+E
+��
+2X −
+1
+σ2(m−1)rX + σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ − σ2
+2
+�2�
+1A1(X)1B4(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2
+�3
+8σ2m+1 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� −σ2m−1r/2
+σ/2−σ2m−1r/2
+e−z2/2 �
+z3σ3−2mξ(r) + . . .
+�
+dz,
+where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.
+Region R4
+Finally, for φ4
+d we have
+A3 ∩ C4 =
+�
+z < σ
+2 − σ2m−1r, σ2mr
+2
+< x ≤ σ2
+2 − σz − σ2mr
+2
+�
+,
+and
+E
+��
+2X − σ2
+2 + σZ
+�
+1A3(X)1C4(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2
+�z3
+6 σ3 + . . .
+�
+dz.
+D.4.2
+Second Moment
+For simplicity, we only consider the case for r ≥ σ−2(m−1), the other case follows analogously.
+Region R1
+For φ1
+d we have
+E
+�
+φ1
+d(X, Z)21A1(X)1B1(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ2m−1r/2
+σ/2
+e−z2/2 �
+z3σ3ξ(r) + . . .
+�
+dz
++
+1
+2
+√
+2π
+� 0
+σ/2−rσ2m−1/2
+e−z2/2 �
+z3σ3ξ(r) + . . .
+�
+dz
++ o(1),
+E
+�
+φ1
+d(X, Z)21A1(X)1B2(X, Z)
+�
+=
+1
+2
+√
+2π
+� 0
+σ/2−σ2m−1r/2
+e−z2/2 �
+z3σ3ξ(r) + . . .
+�
+dz
++
+1
+2
+√
+2π
+� −σ2m−1r/2
+σ/2−σ2m−1r
+e−z2/2 �
+z3σ3ξ(r) + . . .
+�
+dz
++ o(1),
+87
+
+where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.
+Region R2
+For φ2
+d we have
+E
+�
+φ2
+d(X, Z)21A3(X)1C1(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2
+�rz2
+2 σ2m+2 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ/2
+σ/2−σ2m−1r/2
+e−z2/2 �
+−z3σ3 + . . .
+�
+dz,
+E
+�
+φ2
+d(X, Z)21A3(X)1C2(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2 �
+z2σ2m+2 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+σ/2−σ2m−1r
+e−z2/2
+�
+−rz2
+2 σ2m+2 + . . .
+�
+dz.
+Region R3
+For φ3
+d we have
+E
+�
+�
+�
+1
+σ2(m−1)rX − σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ + σ2
+2
+�2�2
+1A1(X)1B3(X, Z)
+�
+�
+=
+1
+2
+√
+2π
+� +∞
+σ2m−1r/2
+e−z2/2
+�rz2
+24 σ2m+2 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� σ2m−1r/2
+σ/2
+e−z2/2
+�rz2
+24 σ2m+2 + . . .
+�
+dz,
+E
+�
+�
+�
+2X −
+1
+σ2(m−1)rX + σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ − σ2
+2
+�2�2
+1A1(X)1B4(X, Z)
+�
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2
+�7rz2
+24 σ2m+2 + . . .
+�
+dz
++
+1
+2
+√
+2π
+� −σ2m−1r/2
+σ/2−σ2m−1r/2
+e−z2/2
+�7rz2
+24 σ2m+2 + . . .
+�
+dz,
+Region R4
+For φ4
+d we have
+E
+��
+2X − σ2
+2 + σZ
+�2
+1A3(X)1C4(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2
+�
+−z3
+3 σ3 + . . .
+�
+dz.
+D.4.3
+Third Moment
+Having established that the only possible scaling is given by α = 1/3, β = m/3 with m ≥ 1, we
+now proceed to bound the third moment of φd in this case. For simplicity, we only consider the
+case for r ≥ 1, the other case follows analogously.
+88
+
+Since m ≥ 1, we find that E
+�
+φ1
+d(X, Z)31R1(X, Z)
+�
+= o(1) as d → ∞ since the limits of integra-
+tion all converge to 0. Then, using H¨older’s inequality for φ2
+d, we have
+E
+�
+φ2
+d(X, Z)3�
+≤ CE
+��σ2
+2 − σZ
+�3
+1A3(X)1C1(X, Z)
+�
++ CE
+��
+2X − σ2
+2 + σZ
+�3
+1A3(X)1C2(X, Z)
+�
++ CE
+�
+�
+�
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX +
+�
+1 −
+1
+σ2(m−1)r
+� �σ2
+2 − σZ
+��2�3
+×1A3(X)1C1∪C2(X, Z)]
+=
+C
+2
+√
+2π
+� σ/2−σ2m−1r/2
+−∞
+e−z2/2
+�
+−rz3
+2 σ5 + ...
+�
+dz
++
+C
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2
+�
+−rz3
+2 σ5 + ...
+�
+dz
++
+C
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2 �
+z3σ5ξ(r) + ...
+�
+dz + o(1),
+where ξ : [0, +∞) → R is a function of r only which might change from one line to the other. For
+89
+
+φ3
+d, we have, using again H¨older’s inequality,
+E
+�
+�
+�
+1
+σ2(m−1)rX − σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ + σ2
+2
+�2�3
+1A1(X)1B3(X, Z)
+�
+�
+≤ CE
+��
+1
+σ2(m−1)rX − σZ
+�3
+1A1(X)1B3(X, Z)
+�
++ CE
+�
+�
+�
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ + σ2
+2
+�2�3
+1A1(X)1B3(X, Z)
+�
+�
+=
+1
+2
+√
+2π
+� +∞
+σ2m−1r/2
+e−z2/2
+�
+−z3r
+2 σ5 + ...
+�
+dz
++
+1
+2
+√
+2π
+� +∞
+σ2m−1r/2
+e−z2/2 �
+z3σ5ξ(r) + ...
+�
+dz + o(1)
+E
+�
+�
+�
+2X −
+1
+σ2(m−1)rX + σZ + Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ − σ2
+2
+�2�3
+1A1(X)1B4(X, Z)
+�
+�
+≤ CE
+��
+2X −
+1
+σ2(m−1)rX + σZ
+�3
+1A1(X)1B4(X, Z)
+�
++ CE
+�
+�
+�
+Z2
+2 −
+1
+2σ2
+�
+1
+σ2(m−1)rX − σZ − σ2
+2
+�2�3
+1A1(X)1B4(X, Z)
+�
+�
+=
+1
+2
+√
+2π
+� +∞
+σ2m−1r/2
+e−z2/2
+�z3r
+2 σ5 + ...
+�
+dz
++
+1
+2
+√
+2π
+� +∞
+σ2m−1r/2
+e−z2/2 �
+z3σ5ξ(r) + ...
+�
+dz + o(1),
+where ξ : [0, +∞) → R is a function of r only which might change from one line to the other.
+Finally, for φ4
+d we have
+E
+��
+2X − σ2
+2 + σZ
+�3
+1A3(X)1C4(X, Z)
+�
+=
+1
+2
+√
+2π
+� σ/2−σ2m−1r
+−∞
+e−z2/2
+�z3
+10σ5 + ...
+�
+dz.
+90
+