On the strong law for arrays and for the bootstrap mean and variance

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 convergence for sums of arrays of random elements

“…Recently, Hu and Taylor [6] proved Chung type SLLN for arrays of rowwise independent random variables. More specifically, let {X ni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let {a n , n ≥ 1} be a sequence of real numbers with 0 < a n ↑ ∞.…”

“…In this paper, we apply de Acosta [4] inequality to obtain Hu and Taylor's [6] result in a general Banach space under the assumption that WLLN holds.…”

On Chung‐Teicher type strong law for arrays of vector‐valued random variables

“…where {a ni , 1 ≤ i ≤ k n , n ≥ 1} and {c ni , 1 ≤ i ≤ k n , n ≥ 1} are suitable arrays of constants (weights) and Ꮾ-valued elements, respectively, and 0 denotes the zero-element in Ꮾ. Hu and Taylor [7] considered SLLN for arrays of row-wise independent random vari-…”