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Entropy Jump and Entropic Central Limit Theorem for Independent Sum

Liuquan Yao, Shuai Yuan

TL;DR

The paper addresses an entropic central limit theorem for sums of independent (not necessarily identically distributed) variables under finite Poincaré-constant constraints. It develops Fisher-information inequalities for convolution in the independent setting via a projection-based approach and connects these to entropy through de Bruijn’s identity, establishing entropy-jump lower bounds and CLT results. Under uniform Poincaré control and mild regularity, it proves convergence in $L^1$ of the sum’s density to the Gaussian density, along with quantitative KL-rate bounds and explicit rates depending on a constant $c$ derived from the Poincaré bounds. The work extends FI-CLT results from the i.i.d. case to independent, non-identically distributed sequences and provides a cohesive framework linking projection lemmas, score-function identities, and OU-semigroup-based entropy analysis. These results offer a rigorous, quantitative path from Fisher-information behavior under convolution to entropic convergence in the independent-sum CLT.

Abstract

It is a manuscript for results about entropic central limit theorem for independent sum under finite Poincaré constant conditions.

Entropy Jump and Entropic Central Limit Theorem for Independent Sum

TL;DR

The paper addresses an entropic central limit theorem for sums of independent (not necessarily identically distributed) variables under finite Poincaré-constant constraints. It develops Fisher-information inequalities for convolution in the independent setting via a projection-based approach and connects these to entropy through de Bruijn’s identity, establishing entropy-jump lower bounds and CLT results. Under uniform Poincaré control and mild regularity, it proves convergence in of the sum’s density to the Gaussian density, along with quantitative KL-rate bounds and explicit rates depending on a constant derived from the Poincaré bounds. The work extends FI-CLT results from the i.i.d. case to independent, non-identically distributed sequences and provides a cohesive framework linking projection lemmas, score-function identities, and OU-semigroup-based entropy analysis. These results offer a rigorous, quantitative path from Fisher-information behavior under convolution to entropic convergence in the independent-sum CLT.

Abstract

It is a manuscript for results about entropic central limit theorem for independent sum under finite Poincaré constant conditions.
Paper Structure (7 sections, 15 theorems, 86 equations)

This paper contains 7 sections, 15 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Fisher Information Inequalities for Convolution
  3. Entropy Jump
  4. Central Limit Theorem for Independent Sum
  5. Proof of Proposition \ref{['convolution of J in independent']}
  6. Preparation
  7. Proof of \ref{['simple result like corollary']}

Key Result

Proposition 2.1

( FI-CLT ) Consider $X_1, X_2$ IID with absolutely continuous densities, variance $\sigma^2$ and restricted Poincar$\acute{\hbox{e}}$ constant $R^*$, then

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1: Carlen1991
  • Theorem 3.2: gap-EJ
  • Definition 3.1
  • Theorem 3.3
  • Theorem 3.4
  • ...and 10 more