Nonlinear renewal theory for Markov random walks
type: Item
instance of: scholarly article
title: Nonlinear renewal theory for Markov random walks
zbMATH Open document ID: 0816.60068
DOI: 10.1016/0304-4149(93)00005-Z
author: Vincent F. Melfi
published in: Stochastic Processes and their Applications
publication date: 1995-07-06T00:00:00Z
review text: The nonlinear renewal theorem, stating the existence of a limit distribution of the overshoot over level \(a\), is proved for a process \(\{Z_ n, n= 0,1,2,\dots\}\) which is close to a Markov random walk \(\{S_ n\}\) in the sense of closeness of conditional finite-dimensional distributions. This condition follows \textit{M. Woodroofe} [Ann. Probab. 18, No. 4, 1790-1805 (1990; Zbl 0717.60102)] rather than the mainstream of nonlinear renewal theory, e.g. \textit{D. Siegmund} [Sequential analysis. Tests and confidence intervals (1985; Zbl 0573.62071)], where it is assumed that \(Z_ n= S_ n+ \xi_ n\), where the sequence \(\{\xi_ n\}\) satisfies a slow change condition. The Markov random walk is assumed to satisfy the conditions of the Markov renewal theorem of \textit{H. Kesten} [Ann. Probab. 2, 355-386 (1974; Zbl 0303.60090)] involving some continuity assumptions, rather than a Harris chain set-up. An application to a machine breakdown problem is given.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60K15
zbMATH DE Number: 713917
zbMATH Keywords: overshoot
zbMATH Keywords: Prokhorov metric
zbMATH Keywords: Markov renewal theory
zbMATH Keywords: machine breakdown
zbMATH Keywords: Harris chain
reviewed by: Søren Asmussen
MaRDI profile type: Publication
cites work: Limit theorems for semi-Markov processes and renewal theory for Markov chains
cites work: Hit-and-Run Algorithms for Generating Multivariate Distributions
cites work: Q5560061
cites work: Q5563135
cites work: Q5822308
cites work: Q3995021
cites work: Q5621834
cites work: Q3226265
cites work: Renewal theory for functionals of a Markov chain with general state space
cites work: A nonlinear renewal theory with applications to sequential analysis. I
cites work: Non–linear renewal theory for lattice random walks
cites work: Nonlinear Markov renewal theory with statistical applications
cites work: General Irreducible Markov Chains and Non-Negative Operators
cites work: Change of Time Scale For Markov Processes
cites work: Q3326564
cites work: Sequential analysis. Tests and confidence intervals
cites work: Boundary crossing probabilities and statistical applications
cites work: Q3948486
cites work: On the nonlinear renewal theorem
cites work: A nonlinear renewal theory
description: scientific article; zbMATH DE number 713917