Abstract
We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector.
Highlights
Introduction and the NotationLet Nd, d ≥ 1, be a d-dimensional lattice
We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector
Let Xn n ∈ Nd be a field of independent random variables
Summary
Let Nd, d ≥ 1, be a d-dimensional lattice. The points of this lattice will be denoted by m m1, . . . , md , n n1, . . . , nd , and so forth. We consider a modified version of “sectorial convergence” in which we say that a field of real numbers an indexed by lattice points in Nd converge in the set W to a ∈ R, and write limW an a if and only if for every ε > 0 the inequality |an − a| < ε holds for all but finite number of n ∈ W. We will be studying necessary and sufficient conditions for the weighted strong law of large numbers WSLLN for short for random fields of independent random variables for a general class of weights defined by Feller in 5 and Jajte in 6 The case of such summability methods in the multi-index setting was considered in 4. The necessary and sufficient conditions for the WSLLN for independent random fields, may be seen as an extension of the previous results of 2, 4
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