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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.

MSC:
 60F15 Strong limit theorems 60G60 Random fields