Convergence of weighted averages of martingales in Banach function spaces
type: Item
instance of: scholarly article
title: Convergence of weighted averages of martingales in Banach function spaces
zbMATH Open document ID: 0956.60038
author: Masato Kikuchi
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-03-07T00:00:00Z
review text: Banach function spaces are Banach spaces of random variables between \(L^\infty\) and \(L^1\) with well-behaved norms. Examples are \(L^p\)-spaces, Orlicz spaces and Lorentz spaces. When \(f= (f_n)\) is a martingale and \((w_n)\) a sequence of positive numbers such that \(W_n= \sum^n_{k=1} w_k\to \infty\), \textit{M. Izumisawa} and \textit{N. Kazamaki} [Tôhoku Math. J., II. Ser. 29, 115-124 (1977; Zbl 0359.60050)] proved that \(f\) converges in \(L^p\) if and only if the weighted averages \(W^{-1}_n \sum^n_{k=1} w_kf_k\) converge in \(L^p\). Under an additional condition, this result is extended to most Banach function spaces. It also works when \(X\) is a weighted \(L^p\)-space with a weight function satisfying the usual \(A_p\) condition.
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 1441386
zbMATH Keywords: martingales
zbMATH Keywords: Banach function spaces
zbMATH Keywords: weighted \(L^p\)-spaces
reviewed by: Dominique Lépingle
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6683
OpenAlex ID: W1965097962
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DOI: 10.1006/JMAA.1999.6683
description: scientific article; zbMATH DE number 1441386