样本均值幂级数的收敛性

CONVERGENCE OF POWER SERIES OF SAMPLE MEANS

摘要: 摘要设X₁,X₂,…为iid.,EX₁=0,0<EX₁²<∞。中心极限定理指出:均值序列\barX_n大体上是以1/n^1/2的速度趋于0。但是,在\barX_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\barX_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_n=1^\infty C_n \sqrtn^-\alpha_n<\infty \Leftrightarrow \sum_n=1^\infty C_n\left|\barX_n\right|^\alpha_n \mid<\infty. a.s.等价。

Abstract: Suppose that X₁,X₂, …are independent identically distributed random variables, EX₁=0, \barX_n=\sum_i=1^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 \sum_n=1^\infty C_n \sqrtn^-\alpha_n<\infty \Leftrightarrow \sum_n=1^\infty C_n\left|\barX_n\right|^\alpha_n \mid<\infty.is 0<EX₁²<∞. 2. Even when EX₁²=∞, from \sum_n=1^\infty C_n \sqrtn^-\alpha_n=\infty it follows follows that \sum_n=1^\infty C_n\left|\barX_n\right|^a_n=\infty, a.s..