Strong Laws of Large Numbers for $r$-Dimensional Arrays of Random Variables

Let $K_r$ be the set of $r$-tuples $\mathbf{k} = (k_1, k_2, \cdots, k_r)$ with positive integers for coordinates $(r \geqq 1)$. Let $\{X_k: \mathbf{k} \in \mathbf{K}_r\}$ be a set of i.i.d. random variables with mean zero, and let $\leqq$ denote the coordinate-wise partial ordering on $K_r$. Set $|\mathbf{k}| = k_k k_2 \cdots k_r$ and define, for $\mathbf{k} \in K_r: S_k = \sum_{j\leqq\mathbf{k}} X_j$. If $\{E_k: \mathbf{k} \in K_r\}$ is a set of events indexed by $K_r$, we say (given $\omega$) "$E_k$ f.o." if $\mathbf{\exists} \mathbf{I}(\omega) \in K_r$ such that $\mathbf{k} \nleqq \mathbf{I}$ implies $\omega \in E_k^c$. We say "$E_k$ a.l." if given any $\mathbf{I} \in K_r, \exists \mathbf{k} \geqq \mathbf{I}$ such that $\omega \in E_k$. We prove: (i) If $E\{|X_k|(\log^+ |X_k|)^{r-1}\} = \infty$, then given any $A > 0, P\{|S_k|/|\mathbf{k}| > A \text{a.1.}\} = 1$. Using martingale techniques, we also give a new proof of the converse result due to Zymund: (ii) If $E\{|X_k| (\log^+ |X_k|)^{r-1}\} < \infty$, then given any $\varepsilon > 0, P\{|S_k|/|\mathbf{k}| < \varepsilon \text{f.o.}\} = 1$. For non-identically distributed independent random variables with mean zero, the usual conditions sufficient for convergence of $S_n/n$ to zero in the linearly ordered case are also sufficient for matrix arrays.