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Renewal theory in a random environment

Published online by Cambridge University Press:  24 October 2008

Laurence A. Baxter  and
Linxiong Li
Laurence A. Baxter
Affiliation:
Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
Linxiong Li
Affiliation:
Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
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Abstract

A random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 179 - 190
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Alsmeyer, G.. Parameter-dependent renewal theorems with applications. Zeitshrift für Operations Research 30 (1986), A111–A134.Google Scholar
[2]Alsmeyer, G.. Second-order approximations for certain stopped sums in extended renewal theory. Advances in Applied Probability 20 (1988), 391–410.CrossRefGoogle Scholar
[3]Bourgin, R. D. and Cogburn, R.. On determining absorption probabilities for Markov chains in random environments. Advances in Applied Probability 13 (1981), 369–387.CrossRefGoogle Scholar
[4]Brox, Th.. A one-dimensional diffusion process in a Wiener medium. Annals of Probability 14 (1986), 1206–1218.CrossRefGoogle Scholar
[5]Chow, Y. S. and Teicher, H.. Probability Theory. (Springer-Verlag, 1978).CrossRefGoogle Scholar
[6]Çinlar, E. and Özekici, S.. Reliability of complex devices in random environments. Probability in the Engineering and Informational Sciences 1 (1987), 97–115.CrossRefGoogle Scholar
[7]Cogburn, R.. Markov chains in random environments: the case of Markovian environments. Annals of Probability 8 (1980), 908–916.CrossRefGoogle Scholar
[8]Cohn, H.. On the growth of the multitype supercritical branching process in a random environment. Annals of Probability 17 (1989), 1118–1123.CrossRefGoogle Scholar
[9]Doob, J. L.. Stochastic Procecses (John Wiley, 1953).Google Scholar
[10]Helm, W. E. and Waldmann, K-H.. Optimal control of arrivals to multiserver queues in a random environment. Journal of Applied Probability 21 (1984), 602–615.CrossRefGoogle Scholar
[11]Keener, R. W.. Renewal theory for Markov chains on the real line. Annals of Probability 10 (1982), 942–954.CrossRefGoogle Scholar
[12]Kijima, M. and Sumita, U.. A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments. Journal of Applied Probability 23 (1986), 71–88.CrossRefGoogle Scholar
[13]Neuts, M. F.. Probability (Allyn and Bacon, 1973).Google Scholar
[14]O'Cinneide, C. A. and Purdue, P.. The M/M/∞ queue in a random environment. Journal of Applied Probability 23 (1986), 175–184.Google Scholar
[15]Shaked, M. and Shantitikumar, J. O.. Some replacement policies in a random environment. Probability in the Engineering and Informational Sciences 3 (1989), 117–134.CrossRefGoogle Scholar
[16]Sinai, Ya. O.. The limiting behavior of a one-dimensional random walk in a random medium. Theory of Probability and its Applications 27 (1982), 256–268.CrossRefGoogle Scholar
[17]Tanny, D.. On multitype branching processes in a random environment. Advances in Applied Probability 13 (1981), 464–497.CrossRefGoogle Scholar
[18]Wick, W. D.. Diffusion in a singular random environment. Annals of Probability 18 (1990), 50–67.CrossRefGoogle Scholar