The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory
type: Item
instance of: scholarly article
title: The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory
zbMATH Open document ID: 0532.60060
DOI: 10.1016/0304-4149(83)90037-6
author: Esa Nummelin
author: Pekka Tuominen
published in: Stochastic Processes and their Applications
publication date: 1983-01-01T00:00:00Z
review text: Let \((X_ n)\) be an aperiodic Harris recurrent Markov chain on a general state space (S,B), where B is countably generated. Further let m be an initial probability measure, \(P^ n(x,A)\) the transition probabilities of \((X_ n)\) and w the corresponding invariant probability measure. The authors study subgeometrical rates of convergence in Orey's theorem, i.e. the cases where \(\| mP^ n-w\| =o(1/y(n))\) with y belonging to a class of sequences which grow slower than \(r^ n\) (for \(r>1)\). These results are complementary to those obtained by the authors [ibid. 12, 187-202 (1982; Zbl 0484.60056)] and by the first author and \textit{R. L. Tweedie} [Ann. Sci. Univ. Clermont 61, Math. 14(1976), 145-154 (1977; Zbl 0356.60009) and Ann. Probab. 6, 404-420 (1978; Zbl 0378.60051)] for geometric rates of convergence. Results are applied to non-singular renewal measures on \({\mathbb{R}}\). If the conditions \(\int tF(dt)>0\), \(\int y(t)(1-F(t))dt<\infty\) are satisfied for the increment distribution F then a probabilistic method is provided to estimate the right tail of the renewal measure.
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 3844763
zbMATH Keywords: Orey's theorem
zbMATH Keywords: general ergodicity
zbMATH Keywords: general recurrence
zbMATH Keywords: geometric - subgeometric rate of convergence
zbMATH Keywords: non-singular renewal processes
reviewed by: F. Thomas Bruss
MaRDI profile type: Publication
full work available at URL: 0304-4149(83)90037-6
OpenAlex ID: W2086207285
cites work: Uniform limit theorems for non-singular renewal and Markov renewal processes
cites work: A New Approach to the Limit Theory of Recurrent Markov Chains
cites work: Renewal theory - A Markov process approach
cites work: Q5624436
cites work: On coupling of discrete renewal processes
cites work: Q4072611
cites work: A refinement of the coupling method in renewal theory
cites work: A splitting technique for Harris recurrent Markov chains
cites work: Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory
cites work: Geometric ergodicity and R-positivity for general Markov chains
cites work: Q4048325
cites work: Uniform rates of convergence for Markov chain transition probabilities
cites work: On Absolutely Continuous Components and Renewal Theory
cites work: One-sided error estimates in renewal theory
description: scientific article; zbMATH DE number 3844763