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.
作者
胡泽春
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)
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)。
关键词
分布
测度反集中
Distribution
Measure anti-concentration
作者简介
Corresponding author:胡泽春,Professor,PhD,Email:zchu@scu.edu.cn。
