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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.

Complete convergence under special hypotheses

Stoica G.

Journal of Mathematics and Applications

2012

Vol. 35

91-95

We prove Baum-Katz type theorems along subsequences of random variables under Komlós-Saks and Mazur-Orlicz type boundedness hypotheses.

Convergence rates in the law of large numbers for arrays

Kruglov V., Volodina A.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

63--76

In this paper we present new sufficient conditions for complete convergence for sums of arrays of rowwise independent random variables. These conditions appear to be necessary and sufficient in the case of partial sums of independent identically distributed random variables. Many known results on complete convergence can be obtained as corollaries to theorems proved in this paper.

Complete exact laws

Adler A.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

215--222

Consider independent and identically distributed random variables {X,Xn, n ≥ 1} with xP{X > x} ~α(log x)α, where α > −1 and P{X < −x} = o(P{X > x}). Even though the mean does not exist, we establish Laws of Large Numbers of the form [formula].. for all ε > 0 and a particular nonsummable sequence {cn, n ≥ 1}, where L ≠ 0.