样本均值幂级数的收敛性  

CONVERGENCE OF POWER SERIES OF SAMPLE MEANS

作　　者：白志东[1] 吴月华[2] 陈希孺[1] 

机构地区：[1]中国科技大学 [2]东南大学

出　　处：《应用概率统计》1990年第1期47-56,共10页Chinese Journal of Applied Probability and Statistics

摘　　要：摘要设X_1,X_2,…为iid.,EX_1=0,0<EX_1~2<∞。中心极限定理指出:均值序列{X_n}大体上是以1/n^(1/2)的速度趋于0。但是,在{X_n}中也存在快于或慢于这个收敛速度的子序列。本文的结果从一个意义上显示:这种子序列的存在对{X_n}的整体影响不大。具体地,若{C_n≥0,n=1,2,…}和{α_n≥0,n=1,2,…}为两个常数列且至少有一个为有界,则sum from n=1 to ∞ C_n(n^(1/2)^(-a_n)<∞与 sum from n=1 to ∞ C_n|_n|a_n<∞ a.s.等价。Suppose that X_1, X_2, …are independent identically distributed random variables, EX_1=0, X_n= sum from i=1 to n X_i/n. Let {C_n, n≥1} and {α_n, n≥1} be two sequences of non-negative numbers, at least one of which is bounded. It is proved that 1. The necessary and sufficient condition for is 0<EX_1~2<∞. 2. Even when EX_1~2=∞, from sum from n=1 to ∞ C_n n^(1/2)^(-an)=∞ it follows follows that sum from n=1 to ∞ C_n|_n|^(a_n)=∞, a. s.

关 键 词：样本均值 幂级数 收敛性 极限定理 

分 类 号：O211.4[理学—概率论与数理统计]