Manuel Ordóñez Cabrera

Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights

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 some previous results for weighted sums with nonrandom weights, and one of them is used to obtain a result of convergence for sums with a random number of addends. Furthermore, a result of almost everywhere convergence of the sequence of certain conditional expectations of the row sums is obtained.

On the weak law for randomly indexed partial sums for arrays of random elements in martingale type p Banach spaces

For randomly indexed sums of the form N n i=1 (Xni cni)/bn, where{Xni,i 1,n 1} are random variables,{Nn,n 1} are positive integer-valued random variables,{cni,i 1,n 1} are suitable conditional expectations and{bn,n 1} are positive constants, we establish a general weak law of large numbers. Our result improves that of Hong (3).

On complete convergence for arrays of rowwise independent random elements

A complete convergence theorem for arrays of rowwise independent random variables was proved by Sung, Volodin, and Hu [14]. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.

Convergence rate of the dependent bootstrapped means

In this paper, a Baum--Katz, Erdos, Hsu--Robbins, Spitzer type complete convergence result is obtained for the dependent bootstrapped means.

L 1-Convergence for Weighted Sums of Some Dependent Random Variables

In this article, the authors discuss the L 1-convergence for weighted sums of some dependent random variables under the condition of h-integrability with respect to an array of weights. The dependence structure of the random variables includes pairwise lower case negative dependence and conditions on the mixing coefficient, the maximal correlation coefficient, or the ρ*-mixing coefficient. They prove that all the weighted sums have similar limiting behaviour.

Almost Sure lim sup Behavior of Dependent Bootstrap Means

Abstract In this article, the upper bound for the exact convergence rate (i.e., the law of the logarithm type result) is obtained for dependent bootstrap means.

On complete convergence of weighted sums of random elements

Let {X ni ,1≤i≤k n , n≥1} be an array of rowwise independent B-valued random elements, and {a ni ,1≤i≤k n , n≥1} an array of constants. Under some conditions of Chung and Hu and Taylor types for the arrays, it is shown the equivalence between the convergence of ∑ i=1 k n a ni X ni to zero in L 1, in probability, almost surely and completely.