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].
We prove stability estimates for the Shannon-Stam inequality (also known as the entropypower inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X, Y ∈ R d , the deficit in the Shannon-Stam inequality is bounded from below by the expressionwhere D (· ||G) denotes the relative entropy with respect to the standard Gaussian and the constant C depends only on the covariance structures and the spectral gaps of X and Y . In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.Our approach is based on ideas somewhat related to the ones which appear in [8]: the very highlevel plan of the proof is to embed the variables X, Y as the terminal points of some martingales and express the entropies of X, Y and X+Y as functions of the associates quadratic co-variation processes. One of the main benefits in using such an embedding is that the co-variation process of X + Y can be easily expressed in terms on the ones of X, Y , as demonstrated below. In [8]
We study the a central limit theorem for sums of independent tensor powers, $\frac{1}{\sqrt{d}}\sum \limits _{i=1}^d X_i^{\otimes p}$. We focus on the high-dimensional regime where $X_i \in{\mathbb{R}}^n$ and $n$ may scale with $d$. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if $n^{2p-1}\ll d$, then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a novel application of optimal transport to Stein’s method, which accounts for the low-dimensional structure, which is inherent in $X_i^{\otimes p}$.
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