On necessary and sufficient conditions for the self-normalized central limit theorem In Honor of Professor Chuanrong Lu on His 85th Birthday  

作　　者：Qiman Shao 

机构地区：[1]Department of Statistics, The Chinese University of Hong Kong

出　　处：《Science China Mathematics》2018年第10期1741-1748,共8页中国科学：数学（英文版）

基　　金：supported by Hong Kong Research Grants Council General Research Fund(Grant Nos.14302515 and 14304917)

摘　　要：Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n^2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary.Let X1, X2,... be a sequence of independent random variables and Sn=sum X1 from i=1 to n and Vn2=sum X12 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum Sn/Vn converges to a standard normal distribution if and only if max1≤i≤n｜Xi｜/Vn → 0 in probability and the mean of X1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n｜Xi｜/Vn → 0 in probability, then these sufficient conditions are necessary.

关 键 词：central limit theorem SELF-NORMALIZED independent random variables 

分 类 号：O211.4[理学—概率论与数理统计]