Strong law of large numbers for ``subsequences'' on the plain
type: Item
instance of: scholarly article
zbMATH Open document ID: 0988.60015
publication date: 2002-03-13T00:00:00Z
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1718883
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: random fields
author: O. I. Klesov
MaRDI profile type: Publication
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title: Strong law of large numbers for ``subsequences'' on the plain
published in: Theory of Stochastic Processes
review text: 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.
reviewed by: M. P. Moklyachuk
description: scientific article; zbMATH DE number 1718883