A Strong Law for Weighted Averages of Independent, Identically Distributed Random Variables with Arbitrarily Heavy Tails

Abstract

Let $X_1, X_2, \cdots$ be independent, identically distributed, nondegenerate random variables, let $w_k$ be a sequence of positive numbers and for $n = 1,2, \cdots$ let $S_n = \sum^n_{k=1} w_kX_k$ and $W_n = \sum^n_{k=1} w_k$. The weak (strong) law is said to hold for $\{X_k, w_k\}$ if and only if $S_n/W_n$ converges in probability (almost surely) to a constant. Jamison, Orey and Pruitt (1965) (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 40-44) studied conditions related to these laws of large numbers. In considering the strong law, only distributions with finite first moments are discussed. However, Theorem 2 of this paper shows that a sequence of random variables and a sequence of weights can be chosen so that the strong law holds and so that the random variables have arbitrarily heavy tails. This result also answers some interesting questions concerning the weak law.