Convergence in the \(p\)th-mean and some weak laws of large numbers for weighted sums of random elements in separable normed linear spaces
type: Item
instance of: scholarly article
title: Convergence in the \(p\)th-mean and some weak laws of large numbers for weighted sums of random elements in separable normed linear spaces
zbMATH Open document ID: 0546.60006
DOI: 10.1016/0047-259X(84)90071-X
author: Xiang Chen Wang
author: M. Bhaskara Rao
published in: Journal of Multivariate Analysis
publication date: 1984-01-01T00:00:00Z
review text: Let \(X_ n\), \(n\geq 1\), be a sequence of random elements taking values in a separable Banach space B and \(a_{nk}\), \(n\geq 1\), \(k\geq 1\), a double array of real numbers satisfying \(\sum_{k}| a_{nk}|\leq \Gamma\) for every \(n\geq 1\) for some positive constant \(\Gamma\). The main result of the paper under review can be described as follows. If \(\| X_ n\|\), \(n\geq 1\), is uniformly integrable, then \(\sum_{k}a_{nk}X_ k\), \(n\geq 1\), converges to 0 in probability if and only if \(\sum_{k}a_{nk}f(X_ k)\), \(n\geq 1\), converges to 0 in probability for every f in the dual space \(B^*\).
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3870289
zbMATH Keywords: weak law of large numbers
zbMATH Keywords: uniformly tight random elements
reviewed by: Vakhtang V. Kvaratskhelia
MaRDI profile type: Publication
cites work: Q4114551
cites work: Q3914124
cites work: Q5723419
cites work: Q3843987
cites work: Laws of large numbers for normed linear spaces and certain Frechet spaces
cites work: Q4056770
cites work: Q5609799
cites work: Q5727879
cites work: Weak Laws of Large Numbers in Normed Linear Spaces
cites work: Stochastic convergence of weighted sums in normed linear spaces
cites work: Q4113164
cites work: Laws of large numbers for tight random elements in normed linear spaces
cites work: Convergence of weighted sums of tight random elements
full work available at URL: 0047-259x(84)90071-x
OpenAlex ID: W2039846809
description: scientific article; zbMATH DE number 3870289