Journal of the
Korean Mathematical Society JKMS

Chatterji strengthened version of a theorem for martingales which is a generalization of a theorem of Marcinkiewicz proving that if $X_n$ is a sequence of independent, identically distributed random variables with $E|X_n|^p < \infty, 0 < p < 2$ and $EX_1 = 0$ if $1 \leq p < 2$ then $n^{- 1/p} \sum_{i=1}^n X_i \to 0$ a.s. and in $L^p$. In this paper, we prove a version of law of large numbers for double arrays. If $\{ X_{ij} \}$ is a double sequence of random variables with $E{|X_{11}|}^{p} \log^{+} {|X_{11}|}^{p} < \infty, 0 < p < 2$, then $\lim_{m \lor n \to \infty} {{{\sum_{i=1}^m} \sum_{j=1}^{n} (X_{ij} - a_{ij}) } \over {{(mn)}^{1 \over p}} }= 0 $~~ a.s. and in $L^p$, where $a_{ij}=0$ if $0 < p < 1$, and $a_{ij}=E[X_{ij} |{\mathcal F}_{ij}]$ if $1 \leq p \leq 2$, which is a generalization of Etemadi's Marcinkiewicz-type SLLN for double arrays. This also generalize earlier results of Smythe, and Gut for double arrays of i.i.d. r.v's.

Keywords: strong law of large numbers, double arrays, $L^p$ convergence

MSC numbers: 60G50, 60F15