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Uniform limit theorems for non-singular renewal and Markov renewal processes

Published online by Cambridge University Press:  14 July 2016

Elja Arjas ,
Esa Nummelin  and
Richard L. Tweedie
Elja Arjas
Affiliation:
University of Oulu
Esa Nummelin
Affiliation:
Helsinki University of Technology
Richard L. Tweedie
Affiliation:
C.S.I.R.O. Division of Mathematics and Statistics Canberra
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Abstract

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Keywords

RENEWAL THEORYKEY RENEWAL THEOREMSFORWARD RECURRENCE TIMESRECURRENT MARKOV CHAINSMARKOV RENEWAL PROCESSESSEMI-MARKOV PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 1 , March 1978 , pp. 112 - 125
Copyright
Copyright © Applied Probability Trust 1978 

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