AI 中文总结
本文在局部紧群上引入熵对称化抗性概念，证明非对称伯努利随机变量具有该抗性并给出多维推广，还明确了有限群上此类抗性分布的边界性质及$\text{Z}_3$、$\text{Z}_4$上的对应分布类别。
AI 中文摘要
实数域上的非对称随机变量$X$若满足：所有能使和$X+Y$为对称随机变量的独立实数随机变量$Y$，其方差均大于$X$的方差，则称$X$具有方差对称化抗性。Kagan、Mallows、Shepp、Vanderbei与Vardi（1999）已证明非对称伯努利随机变量具有方差对称化抗性，Pal（2008）利用随机微积分给出了该结论的证明。我们引入局部紧群上的熵对称化抗性概念，即任意独立对称化子$Y$的熵必须大于$X$的熵。我们证明非对称伯努利随机变量具有熵对称化抗性，并给出了其多维推广。此外，我们还探讨了紧群上熵对称化抗性问题的基本性质，特别证明了有限群上任何具有熵对称化抗性的分布必位于概率单纯形的边界上，并精确描述了$\boldsymbol{\text{Z}}_3$与$\boldsymbol{\text{Z}}_4$上所有熵对称化抗性分布的类别。
英文摘要
An asymmetric random variable $X$ in the reals is said to be variance symmetrization resistant if every independent random variable $Y$ in the reals that produces a symmetric sum $X+Y$ has a greater variance than that of $X$. Asymmetric Bernoulli random variables were shown to be variance symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. We introduce the notion of entropic symmetrization resistance on locally compact groups-- this means that the entropy of any independent symmetrizer $Y$ must exceed that of $X$. We show that asymmetric Bernoulli random variables exhibit entropic symmetrization resistance, and show a multidimensional generalization. We also explore basic aspects of the entropic symmetrization resistance problem in compact groups. In particular, we show that any distribution on a finite group that is entropic symmetrization resistant must lie on the boundary of the probability simplex, and describe precisely the class of all entropic symmetrization resistant distributions on $\mathbb{Z}_3$ and $\mathbb{Z}_4$.
Comments32 pages, 3 figures
