非线性概率下行内负相协随机变量阵列的大数定律  

Lawsof Large Numbers for Arrays of Rowwise Negatively Associated Random Variables under Nonlinear Probabilities

作　　者：兰玉婷 冯新伟 张宁 Yu Ting LAN;Xin Wei FENG;Ning ZHANG(School of Statistics and Management,Shanghai University of Finance and Economics,Shanghai 200433,P.R.China;Zhongtai Securities Institute for Financial Studies,Shandong University,Jinan 250100,P.R.China;School of Data Science,Chinese University of Hong Kong(Shenzhen),Shenzhen 518172,P.R.China)

机构地区：[1]上海财经大学统计与管理学院,上海200433 [2]山东大学中泰证券金融研究院,济南250100 [3]香港中文大学(深圳)数据科学学院,深圳518172

出　　处：《数学学报（中文版）》2022年第6期1105-1122,共18页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(12001317);上海市浦江人才计划资助项目(21PJC048);山东省自然科学基金资助项目(ZR2020QA019);山东大学齐鲁青年学者资助项目。

摘　　要：本文首先在上概率空间给出行内负相协随机变量列阵的大数定律,其可以涵盖Kolmogorov型大数定律与Marcinkiewicz-Zygmund型大数定律,并且该随机变量负相协定义弱于非线性概率下现存的部分独立性概念.此外,本文给出了关于行内独立随机变量阵列的强大数定律.以其涵盖的Kolmogorov型大数定律为例,该定理不仅表明样本均值拟必然落于由随机变量上期望与下期望构成的闭区间之中,还证明了在非线性概率下,该区间是拟必然涵盖样本均值的最小区间.同时,该定理还表明样本均值的上、下极限分别收敛到上期望与下期望这两个端点值的上概率均为1.In this paper,a strong law of large numbers for arrays of rowwise negatively associated random variables is obtained under nonlinear probabilities,from which Kolmogorov type and Marcinkiewicz-Zygmund type strong laws of large numbers are derived.And the notion of negative association is weaker than some existing notions of dependence in nonlinear probabilities.Furthermore,an extension of strong law of large numbers for arrays of rowwise independent random variables under nonlinear probabilities is obtained.As a special case,a Kolmogorov type strong law indicates that not only the cluster points of empirical averages lie in the interval between the lower expectation and upper expectation quasi-surely,but such an interval is also the smallest one that covers the empirical averages quasi-surely.Furthermore,the strong law also states that the upper and lower limits of the empirical averages will converge to the upper and lower expectations with upper probabilities one,respectively.

关 键 词：非线性概率 随机变量阵列 负相协随机变量 独立随机变量 大数定律 

分 类 号：O211.4[理学—概率论与数理统计]