独立随机变量的完全收敛性  

THE COMPLETE CONVERGENCE FOR PARTIAL SUMS OF INDEPENDENT RANDOM VARIABLES

作　　者：于浩[1] 

机构地区：[1]杭州大学

出　　处：《应用概率统计》1989年第2期105-116,共12页Chinese Journal of Applied Probability and Statistics

基　　金：国家自然资金资助项目

摘　　要：本文借助于独立随机变量和a.s.收敛与依概率收敛等价性质,将Katz和Baum有关独立同分布随机变量和完全收敛性的许多结果推广到独立不同分布情形。由此还得到独立不同分布随机变量随机下标和的完全收敛性。In this paper, by making use of the equivalenoe between the a. s, eonvergenoe and the convergence in probability for the partial sums of independent random variables, we improved and strengthened a series of Katz and Baum's results to the case that the random variables need not required to be iid. On the above basis, we got results about the complete convergenoe for the randomly seleoted partial sums of independent random variables. Our main result is the following: Theorem 1. Let X_1,X_2,… be a sequence of independent random variables r>1, 0 <t<2, and l(x)>0 be a slowly variable funetion as x→∞. If we have (ⅰ) sum from k=1 to n E|X_k—b_k|^(r't)= 0 (n^(1+α)) for some r'>1, where b_k=EX_k·I (r't>1)+O·I (r't≤1) and 0≤a<r'(1+t/2)·I(r't>2)+(r'-1)·I(r't≤2), (I(A)denotes the indioator function of the set A.) (ⅱ) sum from n=1 to ∞ E|X_n-b_n|^(t(r-1))l(|X_n-b_n|~t)I(X_n-b_n|≥ε·n^(1/t))<+∞ for any ε>0, then we obtain that (ⅲ) sum from n=1 to ∞ n^(r-2)l(n)P((?)| sum from i=1 to k (X_i-b_i)≥ε·n^(l/t))<+∞ for any ε>0 and (ⅳ) sum from n=1 to ∞ n^(r-2)l(n)P((?)| (| sum from i=1 to k (X_i-b_i)|/k^(1/t))≥ε)<+∞ for any ε>0. Conversely, if (ⅲ) or (ⅳ) holds for some sequemee {b_n}, then (ⅱ) holds.

关 键 词：独立随机变量 非随机足标和 随机足标和 完全收敛性 

分 类 号：O211.5[理学—概率论与数理统计]