A complete convergence theorem for arrays of rowwise independent random variables was proposed by Hu et al. (Statist. Probab. Lett. 38 (1998) 27). Two years later, Hu and Volodin (Statist. Probab. Lett. 47 (2000) 209) imposed one additional condition in addendum to the paper. In this paper, we...

For a sequence of Banach space valued random elements {Vn,n[greater-or-equal, slanted]1} (which are not necessarily independent) with the series [summation operator]n=1[infinity] Vn converging unconditionally in probability and an infinite array a={ani, i[greater-or-equal, slanted]n,...

Some notions of uniform integrability of an array of random elements in a separable Banach space with respect to an array of random variables are introduced and characterized, in order to obtain weak laws of large numbers for randomly weighted sums. The paper contains results which generalize...

A classical theorem on the growth of partial sums of independent identically distributed random variables with infinite expectations due to Feller [Feller, W., 1946. A limit theorem for random variables with infinite moments. Amer. J. Math. 66 (2), 257-262] is reinforced and generalized to the...

Sharpened versions of a Kolmogorov's inequality for sums of independent Bernoulli random variables are proved.

In 1976 the author of this note provided a characterization of the Poisson distribution in the set of infinitely divisible distributions. We give another characterization of the Poisson distribution in the set of infinitely divisible distributions.

The notion of conditional compactly uniform pth-order integrability of an array of random elements in a separable Banach space concerning an array of random variables and relative to a sequence of [sigma]-algebras is introduced and characterized. We state a conditional law for randomly weighted...

In this paper we continue our investigation connected with the new approach developed in Ahmed et al. [Ahmed, S.E., Saleh, A.K.Md.E., Volodin, A., Volodin, I., 2006. Asymptotic expansion of the coverage probability of James-Stein estimators. Theory Probab. Appl. 51 (4) 1-14] for asymptotic...

For weighted randomly indexed sums of the form [summation operator]j=1Nnanj(Vnj-cnj) where {anj,j[greater-or-equal, slanted]1,n[greater-or-equal, slanted]1} are constants, {Vnj,j[greater-or-equal, slanted]1,n[greater-or-equal, slanted]1} are random elements in a real separable martingale type p...

For a sequence of lower negatively dependent nonnegative random variables Xn,n[greater-or-equal, slanted]1 , conditions are provided under which almost surely where bn,n[greater-or-equal, slanted]1 is a nondecreasing sequence of positive constants. The results are new even when they are...