Entropy power inequalities in compact groups

2026-08-23

AI Summary

研究了紧致阿贝尔群中随机变量和的熵幂不等式，通过推广离散熵不等式和调和分析估计方法，解决了等号成立条件及稳定性估计问题。

Abstract

Suppose $X,Y$ are independent random variables with values in a compact abelian group $(G,+)$. We examine the following two entropy power-type inequalities: $h(X+Y)\geq \frac{1}{2}h(X)+\frac{1}{2}h(Y)$ and $h(X+Y)\geq \max\{h(X),h(Y)\}$, where the entropy $h(Z)$ of a $G$-valued random variable $Z$ is defined in terms of its density with respect to Haar measure on $G$. For groups that are either connected or finite with no nontrivial subgroups, we precisely characterize the cases of equality and establish explicit, quantitative stability estimates in terms of relative entropy for these two inequalities. The main tools are a generalization of an entropic inequality obtained by Green, Manners and Tao (2023) for discrete entropy, and a harmonic-analytic estimate for the chi-squared contraction coefficient in connected compact groups. As an application, we derive exponential convergence rates to the uniform distribution in relative entropy for random walks on connected compact abelian groups.

Problem

• compact abelian group
• entropy power inequality
• random variables

Innovation

• entropy power inequality
• compact abelian group
• quantitative stability estimate
• chi-squared contraction coefficient