Complete convergence for sums of arrays of random elements

Abstract

Abstract. Let {X ni } be an array of rowwise independent B-valued random elements and {a n } constants such that 0 < a n ↑ ∞. Under some moment conditions for the array, it is shown that n i=1 X ni /a n converges to 0 completely if and only if n i=1 X ni /a n converges to 0 in probability.

On the rate of convergence of bootstrapped means in a Banach space

“…Interestingly, this result will be applied to establish the strong consistency for bootstrapped means taking values in Banach spaces. More precisely, we present Chung type strong law of large numbers for arrays of rowwise independent random elements under conditions similar to those given by Bozorgnia et al [1]; Hu et al [3]; and Sung [6]. This result is of interest since it holds for an arbitrary real separable Banach space without imposing any geometric conditions.…”

“…Recently, Bozorgnia et al [1], Hu et al [3], and Sung [6] proved Chung's type strong laws of large numbers for arrays of rowwise independent random variables or random elements. We now apply Theorem 2.1 to obtain a similar result in a general real separable Banach space under the assumption that the corresponding weak law of large numbers holds.…”

Complete Moment Convergence and Mean Convergence for Arrays of Rowwise Extended Negatively Dependent Random Variables

“…Sung [ 20 ], Gan and Chen [ 21 ], and Wu and Zhu [ 13 ] extended Theorems A and B to the cases of B -valued random elements, NA random variables, and NOD random variables, respectively. The goal of this paper is to study complete moment convergence and mean convergence for arrays of rowwise END random variables.…”