zbMATH — the first resource for mathematics

A new method for the strong law of large numbers for random fields. (English) Zbl 0941.60056
A field of random variables \(\{ X(\overline n)\), \({\overline n}\in N^{d}\}\) is considered. Let \(S(\overline n)= \sum_{{\overline m}\leq{\overline n}}X(\overline m).\) The author notes that no assumptions on the dependence structure are made. Let \(\{ b(\overline n),{\overline n}\in N^{d}\}\) be a monotone field of positive numbers, that is \(b(\overline m)\leq b(\overline n)\) if \({\overline m} <{\overline n}.\) Let \(c>1\) be a constant and let the set \(\{ A_{t}\}\) be defined by \[ A_{t}=\{ {\overline n}\colon b(\overline n)\leq c^{t}\}. \tag{1} \] The author supposes that there exist a number \(p>0\) and a field of positive numbers \(\{ \lambda(\overline n), {\overline n}\in N^{d}\}\) such that \[ E\left[\max\limits_{{\overline m}\in A_{t}} |S(\overline m)|\right]^{p}\leq \sum\limits_{{\overline m}\in A_{t}}\lambda(\overline m), \quad t>0. \tag{2} \] The following assertion is the main result of this paper: Let \(p>0\). Let a field of random variables \(\{ X(\overline n),{\overline n}\in N^{d}\}\) satisfy condition (2) and let a monotonically increasing field \(\{ b(\overline n),{\overline n}\in N^{d}\}\) of positive numbers satisfy condition (1). If \(\lim_{i}b(\overline n)=\infty\) for \(i=1\) or \(i=d\) and \(\sum_{{\overline n}\in N^{d}} \lambda(\overline n)/ b^{p}(\overline n)<\infty\), then the strong law of large numbers \(\lim_{i}S(\overline n)/ b(\overline n)=0\) a.s. is satisfied. – The author considers the strong law of large numbers for homogeneous random fields too.

60F15 Strong limit theorems
60G60 Random fields