Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean

Abstract

The almost sure limiting behavior of weighted sums of independent and identically distributed random variables barely with or without finite mean are established. Results for these partial sums,

$$\sum\limits_{k = 1}^n {k^\alpha X_k ,} \alpha \in R$$

have been studied, but only when α=−1 or α=0. As it turns out, the two cases of major interest are α=−1 and α>−1. The purpose of this article is to examine the latter.