## Access

You are not currently logged in.

Access JSTOR through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Journal Article

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

R. T. Smythe
The Annals of Probability
Vol. 1, No. 1 (Feb., 1973), pp. 164-170
Stable URL: http://www.jstor.org/stable/2959352
Page Count: 7

#### Select the topics that are inaccurate.

Cancel
Preview not available

## Abstract

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.

• 164
• 165
• 166
• 167
• 168
• 169
• 170