Strong law of large numbers for ``subsequences'' on the plain (Q2777838)

Let \(\{X(m,n);m\geq 1,n\geq 1\}\) be a family of independent and identically distributed random variables and let \(S(m,n)=\sum_{i=1}^m\sum_{j=1}^n X(i,j)\). The strong law of large numbers for the field \(\{X(m,n)\}\) means that \(\lim_{(m,n)\in A,\max\{m,n\}\to\infty}S(m,n)/mn\) exists a.s. The first result of this type goes back to \textit{N. Wiener} [Duke Math. J. 5, 1-18 (1939; Zbl 0021.23501)] who studied the case of a weakly stationary field in the framework of the ergodic theory. \textit{R. T. Smythe} [Ann. Probab. 1, 164-170 (1973; Zbl 0258.60026)] and \textit{A. Gut} [ibid. 11, 569-577 (1983; Zbl 0515.60028)] proposed necessary and sufficient conditions for the strong law of large numbers for the special sets \(A={\mathbb N}^2\) and \(A=\{(m,n)\in{\mathbb N}^2\colon\theta m\leq n\leq\theta^{-1}m,0<\theta<1\}\). NEWLINENEWLINENEWLINEIn this article the necessary conditions \(E|X|<\infty,\sum_{(m,n)\in A}P\{|X|\geq mn\}<\infty\) are proved for the strong law of large numbers with the set \(A=\{(m,n)\in{\mathbb N}^2\colon f(m)\leq n\leq g(m)\}\), where \(f(x),g(x)\) are positive nondecreasing functions such that \(f(x)\leq x\leq g(x)\). These conditions were proposed by the author and \textit{Z. Rychlik} [Theory Probab. Math. Stat. 58, 35-41 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 31-37 (1998; Zbl 0943.60025)]. In this article the author involves more probabilistic arguments and does not use the number theory arguments being the main tool in the previous article. Moreover, the presented method is easy to extend to higher dimensions. Some examples are proposed.