紧群中的熵功率不等式
Entropy power inequalities in compact groups
AI总结：
该研究针对紧阿贝尔群上的独立随机变量，刻画两类熵功率型不等式的等号情形与稳定性估计，推导随机游走收敛到均匀分布的指数速率。
AI中文摘要：
设X、Y是取值于紧阿贝尔群(G,+)的独立随机变量，我们研究以下两个熵功率型不等式：h(X+Y)≥½h(X)+½h(Y)和h(X+Y)≥max{h(X),h(Y)}，其中G值随机变量Z的熵h(Z)根据其关于G上哈尔测度的密度定义。对于连通或有限且无非平凡子群的群，我们精确刻画等号成立的情形，并基于相对熵建立这两个不等式的显式定量稳定性估计。主要工具是Green、Manners和Tao（2023）针对离散熵得到的熵不等式的推广，以及连通紧群中卡方收缩系数的调和分析估计。作为应用，我们推导了连通紧阿贝尔群上随机游走在相对熵下收敛到均匀分布的指数速率。
英文摘要：
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.
