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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 37, Fasc. 1,
pages 185 - 199
 

STRONG LAW OF LARGE NUMBERS FOR RANDOM VARIABLES WITH MULTIDIMENSIONAL INDICES

Agnieszka M. Gdula
Andrzej Krajka

Abstract: Let (Xn,n-∈ V ⊂ ℕ2) be a two-dimensional random field of independent identically distributed random variables indexed by some subset V of lattice ℕ2  . For some sets V the strong law of large numbers

        ∑
   lim   --k∈V,k≤n-Xk-= μ a.s.
n→ ∞,n∈V     |n|

is equivalent to
                ∑
EX1- = μ  and      P [|X1| > |n|] < ∞.
                n∈V

In this paper we characterize such sets V .

2010 AMS Mathematics Subject Classification: Primary: 60F15; Secondary: 60G50, 60G60.

Keywords and phrases: Strong law of large numbers, sums of random fields, multidimensional index.

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