Limiting Behavior of Weighted Sums of Independent Random Variables

Abstract

In this paper, we study weighted sums $\sum^n_{i=1} c_{n-i} X_i$ of i.i.d. zero-mean random variables $X_1, X_2, \cdots$, under the condition that the sequence $(c_n)$ is square summable. It is proved that such weighted sums are, with probability 1, of smaller order than $n^{1/\alpha}$ (respectively $\log n$, etc.) $\operatorname{iff} E|X_1|^\alpha < \infty$ (respectively $Ee^{t|X_1|} < \infty$ for all $t < \infty$, etc.). Certain analogs of the law of the iterated logarithm for such weighted sums are also obtained.