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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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References

[1] Arjas, E., Nummelin, E. and Tweedie, R. L. Semi-Markov processes on a general state space, a-theory and quasi-stationarity. Submitted for publication.Google Scholar
[2] Breiman, L. (1965) Some probabilistic aspects of the renewal theorem. Trans. 4th Prague Conf. on Inf. Theory, Statist. Dec. Functions and Random Processes, 255–261.Google Scholar
[3] Bretagnolle, J. and Dacunha-Castelle, D. (1967) Sur une classe de marches aléatoires. Ann. Inst. H. Poincaré B 3, 403–431.Google Scholar
[4] Çinlar, E. (1974) Periodicity in Markov renewal theory. Adv. Appl. Prob. 6, 61–78.Google Scholar
[5] Cogburn, R. (1975) A uniform theory for sums of Markov chain transition probabilities. Ann. Prob. 3, 191–214.Google Scholar
[6] Feller, W. (1970) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[7] Halmos, P. R. (1950) Measure Theory. Van Nostrand, Princeton.CrossRefGoogle Scholar
[8] Jacod, J. (1971) Théorème de renouvellement et classification pour les chaînes semi-markoviennes. Ann. Inst. H. Poincaré B 7, 83–129.Google Scholar
[9] Jacod, J. (1974) Corrections et compléments à l'article: ‘Théorème de renouvellement et classification pour les chaînes semi-markoviennes’. Ann. Inst. H. Poincaré B 10, 201–209.Google Scholar
[10] Karlin, S. (1955) On the renewal equation. Pacific J. Math. 5, 229–259.Google Scholar
[11] Kesten, H. (1974) Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355–386.Google Scholar
[12] Mcdonald, D. (1975) Renewal theorem and Markov chains. Ann. Inst. H. Poincaré B 11, 187–197.Google Scholar
[13] Mcdonald, D. (1976) On semi-Markov and semi-regenerative processes II. To appear.Google Scholar
[14] Miller, D. R. (1972) Existence of limits in regenerative processes. Ann. Math. Statist. 43, 1275–1282.CrossRefGoogle Scholar
[15] Nummelin, E. (1976) A splitting technique for f-recurrent Markov chains. Preprint, Helsinki University of Technology.Google Scholar
[16] Nummelin, E. (1977) Uniform and ratio limit theorems for Markov renewal and semiregenerative processes on a general state space. Submitted for publication.Google Scholar
[17] Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand-Reinhold, London.Google Scholar
[18] Pitman, J. W. (1974) Uniform rates of convergence of Markov chain transition probabilities. Z. Wahrscheinlichkeitsth. 29, 194–227.CrossRefGoogle Scholar
[19] Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.Google Scholar
[20] Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
[21] Schäl, M. (1971) Über Lösungen einer Erneuerungsgleichung. Abh. Math. Sem. Univ. Hamburg 36, 89–98.Google Scholar
[22] Schäl, M. (1970) Rates of convergence in Markov renewal processes with auxiliary paths. Z. Wahrscheinlichkeitsth. 16, 29–38.Google Scholar
[23] Smith, W. L. (1954) Asymptotic renewal theorems. Proc. R. Soc. Edinburgh A 64, 9–48.Google Scholar
[24] Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar
[25] Smith, W. L. (1960) Remarks on the paper ‘Regenerative stochastic processes’. Proc. R. Soc. London A 256, 296–501.Google Scholar
[26] Stone, C. R. (1966) On absolutely continuous components and renewal theory. Ann. Math. Statist. 37, 271–275.Google Scholar
[27] Tweedie, R. L. (1974) R-theory for Markov chains on a general state space I. Ann. Prob. 2, 840–864.Google Scholar
[28] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.Google Scholar
[29] Tweedie, R. L. (1977) Hitting times of Markov chains, with application to state-dependent queues. Bull. Austral. Math. Soc. 17, 99–107.Google Scholar