The Annals of Probability

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

R. T. Smythe

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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.

Article information

Source
Ann. Probab., Volume 1, Number 1 (1973), 164-170.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176997031

Digital Object Identifier
doi:10.1214/aop/1176997031

Mathematical Reviews number (MathSciNet)
MR346881

Zentralblatt MATH identifier
0258.60026

JSTOR
links.jstor.org

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G45 60F15: Strong theorems

Keywords
Independent identically distributed random variables strong law of large numbers (reversed) martingale Borel-Cantelli lemma

Citation

Smythe, R. T. Strong Laws of Large Numbers for $r$-Dimensional Arrays of Random Variables. Ann. Probab. 1 (1973), no. 1, 164--170. doi:10.1214/aop/1176997031. https://projecteuclid.org/euclid.aop/1176997031


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