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

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 $E\mathrm{\mid }{X}_1{\mathrm{\mid }}^p<\mathrm{\infty }$ for some $1<p<2$, then $(S_n-E{S}_n)\mathrm{/}{n}^{1\mathrm{/}p}\to 0$ a.s. where $S_n={\sum }_{k=1}^n{X}_k$.