Mean Convergence Theorems and Weak Laws of Large Numbers for Double Arrays of Random Elements in Banach Spaces

Abstract

Abstract. For a double array of random elements {V mn ; m ≥ 1, n ≥ 1} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under whichThe weak law results provide con-→ 0 in probability where {Tm; m ≥ 1} and {τn; n ≥ 1} are sequences of positive integer-valued random variables, {kmn; m ≥ 1, n ≥ 1} is an array of positive integers. The sharpness of the results is illustrated by examples.

Strong Laws of Large Numbers for Weighted Sums of Hilbert-valued Coordinatewise PNQD Random Variables with Applications

“…Obviously, if a sequence of -valued random variables is pairwise independent then it is coordinatewise PNQD. In [3], Example 2.3, the authors showed a sequence of -valued coordinatewise PNQD random variable which is not pairwise independent.…”

“…Recently, the results for weak laws of large numbers for weighted coordinatewise PNQD random vectors in Hilbert spaces in the case that the decay 𝑟-th order 02 r of tail probability was established by Dung and Son in [3]. In 2022, Son and Cuong [5] investigated the complete convergence and strong laws of large numbers for weighted sums of coordinatewise PNQD random variables taking values in Hilbert spaces.…”

Convergence of Double Series of Random Elements in Banach Spaces

“…Strong law of larger number for double array of random element in Banach spaces have studied by many authors. For example, Dung et al [1], Dung and Tien [2], Quang et al [8], Roralsky and Thanh [9], Stadtmuller and Thanh [11]. The three-series theorem for martingale in Banach spaces in case of single series was established by Tien [13].…”