发表机构
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究使随机变量对称化所需的随机性，通过指数卷积的微分反演公式及离散差分算子等方法，对指数分布和几何分布建立对称化抗性结果，给出熵和方差对称化抗性分布新例，且表明特定条件下等式成立。
AI 中文摘要
给定一个随机变量\(X\)，若独立随机变量\(Y\)满足\(X + Y\)的分布关于原点对称，则称\(Y\)是\(X\)的对称化变量。对称化抗性研究关注是否每个这样的\(Y\)都必须包含至少与\(X\)一样多的随机性。此前针对二元随机变量从方差和香农熵角度进行过研究。本文针对指数分布和几何分布建立了精确的对称化抗性结果。对于指数随机变量\(X\)，证明了每个绝对连续独立对称化变量\(Y\)满足\(h_\alpha (Y)\ge h_\alpha(X)\)（\(h_\alpha(\cdot)\)是\(\alpha>0\)阶的Rényi熵），以及Tsallis熵和方差的类似不等式。证明基于指数卷积的微分反演公式，将对称化约束转化为风险率不等式。利用相应的离散差分算子和离散尾部比较论证，得到了几何随机变量的平行结果。此外，表明当且仅当\(Y\)是\(-X\)的独立副本时等式成立。我们的结果提供了熵和方差对称化抗性分布的新例子。
英文摘要
Given a random variable $X$, an independent random variable $Y$ is called a symmetrizer of $X$ if their sum $X+Y$ is symmetric about the origin. The study of symmetrization resistance asks whether every such $Y$ must contain at least as much randomness as $X$. This problem was previously investigated for binary random variables in terms of variance and Shannon entropy. In this paper, we establish sharp symmetrization resistance results for exponential and geometric distributions. We prove that every independent symmetrizer $Y$ of an exponential random variable $X$ has variance, Rényi and Tsallis entropies of every positive order at least as large as that of $X$. Parallel results are obtained for integer-valued independent symmetrizers of geometric random variables. Equality in each comparison holds precisely when $Y$ is an independent copy of $-X$. Furthermore, we show the majorization of $X$ over $Y$ via establishing sharp concentration inequalities for $Y$. The proofs crucially rely on a differential/difference inversion formula for exponential/geometric convolution, which converts the symmetrization constraint into a hazard-rate inequality and enables us to identity exponential/geometric distributions as extremal distributions of a corresponding optimization problem.
