两类独立随机变量和的概率估计  

Probability Estimation of the Sum of Two Types of Independent Random Variables

作　　者：叶柳 马丽[1] YE Liu;MA Li(School of Mathematics and Statistics/Key Laboratory of Data Science and Smart Education,Ministry of Education,Hainan Normal University,Haihou 571158,China)

机构地区：[1]海南师范大学数学与统计学院/数据科学与智慧教育教育部重点实验室

出　　处：《海南师范大学学报（自然科学版）》2022年第2期175-178,199,共5页Journal of Hainan Normal University（Natural Science）

基　　金：国家自然科学基金项目(11861029);海南省高层次人才项目(120RC589);海南省研究生创新科研课题(Qhys2021-301);海南省自然科学基金面上项目(122MS056)。

摘　　要：相互独立随机变量和的概率估计是概率统计中一个重要的研究方向。本文研究了两类独立随机变量和的概率估计:一类是n个相互独立的服从{-1,1}上的均匀分布的随机变量和Sn的最大值的概率估计,另一类是两个独立随机变量和的概率估计。首先,用全概率公式、递推的方法及随机变量的对称性给出了log P(max_(1≤i≤n)|S_(i)≤C|)的表达式,其中C为常数且1≤C <2;对一般的C≥1,通过对偶数项和奇数项进行分类讨论,用全概率公式和递推的方法得到了该对数的下界。其次,对两个独立的随机变量,本文证明了如果其分布函数的对数大于等于幂函数,则这两个随机变量的和的分布函数的对数也大于等于某个幂函数。The probability estimation of the sum of mutually independent random variables is an important research direction in probability and statistics. In this paper, the probability estimations of two types of independent random variables’ sum are studied: the first one is the probability estimation of the maximum sum of n independent random variables that are uniformly distributed on {-1, 1};the second one is the probability estimation of the sum of two independent random variables. Firstly, by using the total probability formula, recursive method and the symmetry of random variables, the expression of log P(max_(1≤i≤n)|S_(i)≤C|) is given, where C is a constant greater than or equal to 1 and less than 2. For general C ≥ 1, the low bound of log P(max_(1≤i≤n)|S_(i)≤C|) is also got by considering whether n is even or not. Secondly, about the two independent random variables, this paper proves that if the logarithm of each random variable’s distribution is greater than or equal to some power functions, then the logarithm of their sum’s distribution is also greater than or equal to some power functions.

关 键 词：简单随机游动 独立随机变量和的分布 概率估计 

分 类 号：O211.1[理学—概率论与数理统计]