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On transient regenerative processes

Published online by Cambridge University Press:  14 July 2016

Emily Murphree  and
Walter L. Smith
Emily Murphree*
Affiliation:
Miami University
Walter L. Smith*
Affiliation:
University of North Carolina
*
∗Postal address: Department of Mathematics and Statistics, Miami University, Bachelor Hall, Oxford, OH 45056, USA.
∗∗Postal address: Department of Statistics, University of North Carolina, Chapel Hill, NC 27514, USA.
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Abstract

A transient cumulative process based upon a sequence of possibly infinite lifetimes is defined, and examples of such a process are described. Given a mild condition on the improper lifetime distribution and given that all lifetimes observed by time t are finite, the expected value of this transient process at t is related to the expected value of a cumulative process based upon proper lifetimes. This relationship is exploited to show that the conditional behavior of the transient process is analogous to that of a proper process and, in particular, the transient process is asymptotically normal.

Keywords

CUMULATIVE PROCESSESRENEWALSNORMAL LIMIT LAWS
Type
Research Papers
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 52 - 70
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research supported by the Office of Naval Research under grant N00014–76-C0550.

References

Bellman, R. and Harris, T. E. (1948) On the theory of age-dependent stochastic branching processes. Proc. Nat. Acad. Sci. USA 34, 601–604.Google Scholar
Bondarenko, O. N. (1960) Age-dependent Branching Processes. Ph.D. Dissertation, Moscow State University.Google Scholar
Chistyakov, V. P. (1964) A theorem on sums of independent positive random variables. Theory Prob. Appl. 9, 640–648.Google Scholar
Cramer, H. (1955) Collective Risk Theory. Jubilee volume of Försäkringsbolaget Skandia, Stockholm.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Kingman, J. F. C. (1972) Regenerative Phenomena. Wiley, New York.Google Scholar
Lamperti, J. (1961) A contribution to renewal theory. Proc. Amer. Math. Soc. 12, 724–731.Google Scholar
Siegmund, D. (1975) The time until ruin in collective risk theory. Mitt. Verein. Schweiz Versich. Math. 75 (2), 157–165.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar
Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 243–302.Google Scholar
Vinogradov, O. P. (1964) On an age-dependent branching process. Theory Prob. Appl. 9, 131–136.Google Scholar
Von Bahr, B. (1974) Ruin probabilities expressed in terms of ladder height distributions. Scand. Actuar. J. , No. 4, 190–204.Google Scholar