Propagation of Localization Optimal Entropy Production and Convergence rates for the Central Limit Theorem

TL;DR

This paper investigates how localization properties propagate under convolution and uses these results to establish optimal rates of entropy production and convergence in the Central Limit Theorem for distributions with finite fourth moments.

Contribution

It introduces a method to propagate various localization types through convolution and applies this to prove the optimal entropy convergence rate in the CLT.

Findings

Propagation of polynomial, exponential, and Gaussian localization under convolution

Optimal entropy production rate of 1/√n in the CLT for finite 4th moment distributions

Convergence in entropy and L^1 sense at the optimal rate

Abstract

We prove for the rescaled convolution map $f\to f\circledast f$ propagation of polynomial, exponential and gaussian localization. The gaussian localization is then used to prove an optimal bound on the rate of entropy production by this map. As an application we prove the convergence of the CLT to be at the optimal rate $1/\sqrt{n}$ in the entropy (and $L^1$) sense, for distributions with finite 4th moment.