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

  • Elja Arjas (a1), Esa Nummelin (a2) and Richard L. Tweedie (a3)
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.

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[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.
[2] Breiman, L. (1965) Some probabilistic aspects of the renewal theorem. Trans. 4th Prague Conf. on Inf. Theory, Statist. Dec. Functions and Random Processes, 255261.
[3] Bretagnolle, J. and Dacunha-Castelle, D. (1967) Sur une classe de marches aléatoires. Ann. Inst. H. Poincaré B 3, 403431.
[4] Çinlar, E. (1974) Periodicity in Markov renewal theory. Adv. Appl. Prob. 6, 6178.
[5] Cogburn, R. (1975) A uniform theory for sums of Markov chain transition probabilities. Ann. Prob. 3, 191214.
[6] Feller, W. (1970) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.
[7] Halmos, P. R. (1950) Measure Theory. Van Nostrand, Princeton.
[8] Jacod, J. (1971) Théorème de renouvellement et classification pour les chaînes semi-markoviennes. Ann. Inst. H. Poincaré B 7, 83129.
[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, 201209.
[10] Karlin, S. (1955) On the renewal equation. Pacific J. Math. 5, 229259.
[11] Kesten, H. (1974) Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355386.
[12] Mcdonald, D. (1975) Renewal theorem and Markov chains. Ann. Inst. H. Poincaré B 11, 187197.
[13] Mcdonald, D. (1976) On semi-Markov and semi-regenerative processes II. To appear.
[14] Miller, D. R. (1972) Existence of limits in regenerative processes. Ann. Math. Statist. 43, 12751282.
[15] Nummelin, E. (1976) A splitting technique for f-recurrent Markov chains. Preprint, Helsinki University of Technology.
[16] Nummelin, E. (1977) Uniform and ratio limit theorems for Markov renewal and semiregenerative processes on a general state space. Submitted for publication.
[17] Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand-Reinhold, London.
[18] Pitman, J. W. (1974) Uniform rates of convergence of Markov chain transition probabilities. Z. Wahrscheinlichkeitsth. 29, 194227.
[19] Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.
[20] Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.
[21] Schäl, M. (1971) Über Lösungen einer Erneuerungsgleichung. Abh. Math. Sem. Univ. Hamburg 36, 8998.
[22] Schäl, M. (1970) Rates of convergence in Markov renewal processes with auxiliary paths. Z. Wahrscheinlichkeitsth. 16, 2938.
[23] Smith, W. L. (1954) Asymptotic renewal theorems. Proc. R. Soc. Edinburgh A 64, 948.
[24] Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 631.
[25] Smith, W. L. (1960) Remarks on the paper ‘Regenerative stochastic processes’. Proc. R. Soc. London A 256, 296501.
[26] Stone, C. R. (1966) On absolutely continuous components and renewal theory. Ann. Math. Statist. 37, 271275.
[27] Tweedie, R. L. (1974) R-theory for Markov chains on a general state space I. Ann. Prob. 2, 840864.
[28] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737771.
[29] Tweedie, R. L. (1977) Hitting times of Markov chains, with application to state-dependent queues. Bull. Austral. Math. Soc. 17, 99107.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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