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 $X_1,X_2,\dots$ are pairwise independent identically distributed random variables such that $E|X_1{|}^p<\infty$ for some $1<p<2$, then $(S_n-E{S}_n)/n^{1/p}\to 0$ a.s. where $S_n={\sum }_{k=1}^n{X}_k$.

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