一些常见分布族的反集中函数  

On the Anticoncentration Functions of Some Familiar Families of Distributions

作　　者：胡泽春[1] 宋仁明 谭渊 Hu Zechun;Song Renming;Tan Yuan(College of Mathematics,Sichuan University,Chengdu 610065,China;Department of Mathematics,University of Illinois UrbanaChampaign,Urbana,IL 61801,USA)

机构地区：[1]四川大学数学学院,成都610065 [2]伊利诺伊大学厄巴纳香槟分校数学系,IL61801

出　　处：《数学理论与应用》2024年第1期1-15,共15页Mathematical Theory and Applications

基　　金：supported by the National Natural Science Foundation of China(Nos.12171335,11931004,12071011);the Science Development Project of Sichuan University(No.2020SCUNL201);the Simons Foundation(No.960480)。

摘　　要：假定{X_(α)}为一族服从某类分布的随机变量,具有有限期望E[X_(α)]和有限方差Var(X_(α)),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→inf_(α)P(|X_(α)-E[X_(α)]|≥y√Var(X_(α))),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.Let{X_(α)}be a family of random variables following a certain type of distributions with finite expectation E[X_(α)]and finite variance Var(X_(α)),whereαis a parameter.Motivated by the recent paper of Hollom and Portier(arXiv:2306.07811v1),we study the anticoncentration function(0,∞)∋y→inf_(α)P(|X_(α)-E[X_(α)]|≥y√Var(X_(α)))and find its explicit expression.We show that,for certain familiar families of distributions,including the uniform,exponential,nondegenerate Gaussian and student’s t-distributions,the anticoncentration function is not identically zero,which means that the corresponding families of random variables have some sort of anticoncentration property;while for some other familiar families of distributions,including the binomial,Poisson,negative binomial,hypergeometric,Gamma,Pareto,Weibull,lognormal and Beta distributions,the anticoncentration function is identically zero.

关 键 词：分布 测度反集中 

分 类 号：O211[理学—概率论与数理统计]