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Strong law of large numbers for random variables with multidimensional indices

Gdula A. M., Krajka A.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 1

185--199

Let {Xn, n ϵ 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 [wzór] is equivalent to EX1 = μ and [wzór]. In this paper we characterize such sets V.

On the complete convergence of randomly weighted sums of random fields

Gdula A. M., Krajka A.

Demonstratio Mathematica

2014

Vol. 47, nr 1

232--252

Let (…) be a d-dimensional random field indexed by some subset V of lattice Nd, which are stochastically dominated by a random variable X. Let (…) be a 2d-dimensional random field independent of (…) and such that (…) for some constant M. In this paper, we give conditions under which the following series (…), is convergent for some real t, some fixed p > 0 and all ε > 0. Here |n| is used for (…). The randomly indexed sums of field (…) are considered too.