The CLT in high dimensions: quantitative bounds via martingale embedding

Abstract

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author [38], for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) We derive the first non-asymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors; (c) We give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed in [19].

Quantitative Weak Convergence for Discrete Stochastic Processes

“…Prior to this, a number of other authors have proved an optimal O(1/ √ k) rate, but without establishing dimension dependence [see, e.g., 1,2,15]. Following [17], [9] improved the rate to O(…”

“…, where C is the Poincare constant. It is worth noting that the β term in [17], [9] and in the results of this paper is typically on the order of √ d, whereas the term C in [6] is typically dimension-free. On the other hand, the assumptions of [17] and [9] are incompatible with [6].…”

“…It is worth noting that the β term in [17], [9] and in the results of this paper is typically on the order of √ d, whereas the term C in [6] is typically dimension-free. On the other hand, the assumptions of [17] and [9] are incompatible with [6]. In [5], under more general assumptions than [17,9] the author proved an optimal √ d dimensional dependence, but with suboptimal k 1/4 dependence.…”

“…On the other hand, the assumptions of [17] and [9] are incompatible with [6]. In [5], under more general assumptions than [17,9] the author proved an optimal √ d dimensional dependence, but with suboptimal k 1/4 dependence.…”

“…Thus, the sequence S k essentially has the same dynamics as x k from (3), with U (x) = − 1 2 x 2 2 , Tη k = η k and variable stepsize δ k = 1 k . Assuming ηi 2 ≤ β almost surely, the fastest convergence result is is proven in Theorem 1.1 of [9], with a rate of…”