Growth of Sums of Pairwise Independent Random Variables with Infinite Expectations

Abstract

It is proved that ${\bf P}\{|S_n|>a_n $ infinitely often$\}=0$ or 1 if the series $\sum_{n=1}^{\infty}{\bf P}\{|X_n|>a_n\}$ is convergent or nonconvergent, where $S_n=X_1+\cdots+X_n$ is a sum of identically distributed pairwise independent random variables with infinite expectations, $a_n>0$, for some m a sequence $\{a_n\}_{n\geq m}$ strictly increasing and convex.