Blackwell's renewal theorem for certain linear submartingales and coupling
type: Item
instance of: scholarly article
title: Blackwell's renewal theorem for certain linear submartingales and coupling
zbMATH Open document ID: 0799.60083
DOI: 10.1007/BF00994262
author: Gerold Alsmeyer
published in: Acta Applicandae Mathematicae
publication date: 1994-11-17T00:00:00Z
review text: The paper is devoted to the extensions of the Blackwell's renewal theorem from the classical case of nonnegative summands \(X_ n\), \(n\geq 1\), to the more general situation. The sequence \((X_ n)\) is assumed to satisfy two main conditions: stochastic boundedness, i.e. \[ G(t)\leq P(X_{n+1}\leq t\mid{\mathcal F}_ n)\leq F(t) \] for some distributions \(F\) and \(G\) with finite means; and stochastic stability, i.e. \[ \lim_{k\to\infty} \sup_{n\geq 0} \biggl\| k^{-1} \sum^ k_{j=1} E(X_{n+j}\mid{\mathcal F}_ n)-\theta \biggr\|_ \infty=0, \] for some \(\theta\in{\mathbb{R}}\), where \(({\mathcal F}_ n)_{n\geq 1}\) is the corresponding filtration and \(\|\cdot\|_ \infty\) denotes the \(L_ \infty\)-norm. Proofs of the results are based on the coupling method. A set of examples is also given.
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 579595
zbMATH Keywords: Blackwell's renewal theorem
zbMATH Keywords: stochastic boundedness
zbMATH Keywords: stochastic stability
zbMATH Keywords: coupling method
reviewed by: Eugene Kozarovitsky
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00994262
OpenAlex ID: W1983720106
description: scientific article; zbMATH DE number 579595