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Quasi-renewal estimates

Part of: Mathematical biology in general Special processes

Published online by Cambridge University Press:  14 July 2016

Didier Piau
Didier Piau*
Affiliation:
Université Lyon-I
*
∗Postal address: Laboratoire de Probabilités, Université Claude Bernard (Lyon-I), 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France. Email address: piau@jonas.univ-lyon1.fr
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Abstract

We show that the solution of a quasi-renewal equation with an exponential distribution of the renewals converges at infinity and we compute explicitly the limit, hence generalizing the classical renewal theorem. We apply this result to a stochastic model of DNA replication introduced by Cowan and Chiu (1994).

Keywords

Renewal equationLaplace transform

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K10: Applications (reliability, demand theory, etc.) 92B05: General biology and biomathematics
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 269 - 275
Copyright
Copyright © 2000 by The Applied Probability Trust 

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References

Andrews, G. (1976). The theory of partitions. In Encyclopedia of Mathematics and its Applications, Vol. 2., Section: Number Theory, Addison-Wesley, Reading, MA.Google Scholar
Cowan, R., and Chiu, S. N. (1994). A stochastic model of fragment formation when DNA replicates. J. Appl. Prob. 31, 301–308.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Lindvall, E. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar