Chung type strong laws for arrays of random elements and bootstrapping

Abstract

Let {Xnk) be an array of rowwise independent random elements in a separable Banach space. Chung type strong laws of large numbers are obtained under various moment conditions on the random elements and geo~netric type p, 1 _< p 5 2, conditions on the Banach space. Comparisons with existing results for arrays of random elements are provided to illustrate the strength of these results. The results can be directly applied to show the asymptotic validity of the bootstrap mean and variance for random functions.

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

“…In this case, the main result of Bozorgnia et al [1] can be seen as a special case of the results given in Theorem 3.2 of this paper.…”

“…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.…”

“…We give an example verifying the conditions of Theorem 3.2 by providing an alternative proof of the main result of Bozorgnia et al [1,Theorem 3.1]. This proof is substantially simpler than the original one.…”

“…Some results on the consistency of the bootstrapped mean of random elements in Banach spaces are given in [1]. However, these consistency results impose a geometric condition (Rademacher type p) on the Banach space.…”

“…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.…”