On the laws of large numbers for double arrays of independent random elements in Banach spaces
type: Item
instance of: scholarly article
title: On the laws of large numbers for double arrays of independent random elements in Banach spaces
zbMATH Open document ID: 1305.60022
author: Andrew Rosalsky
author: Le Van Thanh
author: Nguyen Thi Thuy
publication date: 2014-09-11T00:00:00Z
review text: The paper considers a double array \(\{V_{mn}, m \geq 1, n \geq 1\}\) of independent random elements on a probability space \((\Omega, \mathcal{F},P)\) and taking values in a real separable Banach space \(\chi\) with norm \(||\cdot||\). It provides conditions under which the strong law of large numbers (SLLN) and the weak law of large numbers (WLLN) for the double sum \(\sum_{i=1}^n \sum_{j=1}^m V_{ij}\) are equivalent. Such double sums differ substantially from the the partial sums \(\sum_{i=1}^n V_i,\; n \geq 1\), of a sequence of independent random elements \(\{V_n, n \geq 1\}\). The paper treats both the i.i.d case and the independent but not identically distributed case. The main results are applied to obtain Kolmogorov, Brunk-Chung and Marcinkiewicz-Zygmund type SLLNs for double sums in Rademacher type \(p\) \((1 \leq p \leq 2)\) Banach spaces.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 6342507
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: weak law of large numbers
zbMATH Keywords: real separable Banach space
zbMATH Keywords: almost sure convergence
zbMATH Keywords: convergence in probability
reviewed by: Pedro A. Morettin
MaRDI profile type: Publication
full work available at URL: s10114-014-3507-7
OpenAlex ID: W2069647851
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DOI: 10.1007/S10114-014-3507-7
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6342507