Strong law of large numbers on partially ordered sets.(English. Russian original)Zbl 0943.60025

Theory Probab. Math. Stat. 58, 35-41 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 31-37 (1998).
Let $$\{ X(k,l);k\geq 1, l\geq 1\}$$ be a sequence of independent identically distributed random variables. The authors investigate the sum $$S(m,n)=\sum_{k=1}^{n} \sum_{l=1}^{n}X(k,l).$$ They prove the strong law of large numbers for $$S(m,n).$$ The following theorem is the main result of this paper: Let $$f$$ and $$g$$ be positive functions such that: 1) $$f$$ is a monotone increasing function; 2) $$f(x)\leq x\leq g(x)$$; 3) $$f(x)/x\downarrow$$, $$g(x)/x\uparrow.$$ Let $$A_{f,g}$$ be a sector $$A_{f,g}=\{ m,n\colon f(m) \leq n\leq g(m)\}.$$ Then $\lim_{\substack{ m,n\to\infty\\ (m,n)\in A_{f,g}}} {S(m,n)\over mn}=0 \qquad \text{a.s.}$ if and only if $$EX(1,1)=0$$ and $$\sum_{m,n\in A_{f,g}} P(|X|\geq mn)<\infty.$$

MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks