| 叶 柳，马 丽*.两类独立随机变量和的概率估计[J].海南师范大学学报自科版,2022,35(2):175-178 |
| 两类独立随机变量和的概率估计 |
| Probability Estimation of the Sum of Two Types ofIndependent Random Variables |
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| DOI：10.12051/j.issn.1674-4942.2022.02.012 |
| 中文关键词: 简单随机游动 独立随机变量和的分布 概率估计 |
| 英文关键词: simple random walk the distribution of the sum of independent random variables probability estimation |
| 基金项目: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） |
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| 中文摘要: |
| 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 S n
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 logP ( 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。 |
| 英文摘要: |
| The probability estimation of the sum of mutually independent random variables is an important research direc⁃
tion 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 vari⁃
ables. Firstly, by using the total probability formula, recursive method and the symmetry of random variables, the expression
of logP ( 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 logP ( max
1 ≤ i ≤ n | S i |
≤ C ) is also got by considering whether n is even or not. Secondly, about the two independent ran⁃
dom 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. |
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