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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
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