CONVERGENCE OF POWER SERIES OF SAMPLE MEANS

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..