摘要
假定{Xα}为一族服从某类分布的随机变量,具有有限期望E[Xα]和有限方差Var(Xα),其中α为一参数.受Hollom和Portier的论文(arXiv:2306.07811v1)的启发,在本文中我们考虑反集中函数(0,∞)∋ y →infαP |Xα-E[Xα]|≥y√Var(Xα),并给出其清晰表示.我们将证明,对于某些常见分布族,包括均匀分布、指数分布、非退化高斯分布和学生t-分布,反集中函数不恒为零,这表明相应随机变量族具有某种反集中性质;然而对另外一些常见分布族,包括二项分布、泊松分布、负二项分布、超几何分布、伽马分布、帕雷托分布、威布尔分布、对数正态分布和贝塔分布,反集中函数恒为零.
Abstract
Let{Xα}be a family of random variables following a certain type of distributions with finite expectation E[Xα]and finite variance Var(Xα),where α is a parameter.Motivated by the recent paper of Hollom and Portier(arXiv:2306.07811v1),we study the anti-concentration function(0,∞)∋ y →infα P |Xα-E[Xα]|≥y√Var(Xα)and find its explicit expression.We show that,for certain familiar families of distributions,including the uniform,exponential,non-degenerate Gaussian and student's t-distributions,the anti-concentration function is not identically zero,which means that the corresponding families of random variables have some sort of anti-concentration property;while for some other familiar families of distributions,including the binomial,Poisson,negative binomial,hypergeometric,Gamma,Pareto,Weibull,log-normal and Beta distributions,the anti-concentration function is identically zero.
基金项目
国家自然科学基金(12171335)
国家自然科学基金(11931004)
国家自然科学基金(12071011)
Science Development Project of Sichuan University(2020SCUNL201)
Simons Foundation(960480)
维普
万方数据