Marcinkiewicz-Zygmund Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

Abstract

It is shown that the Marcinkiewicz-Zygmund strong law of large numbers holds for pairwise independent identically distributed random variables. It is proved that if $X_{1}, X_{2}, \ldots$ are pairwise independent identically distributed random variables such that $E|X_{1}|^p < \infty$ for some $1 < p < 2$, then $(S_{n}-ES_{n})/n^{1/p} \to 0$ a.s. where $S_{n} = \sum_{k=1}^{n} X_{k}$