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Finite-size corrections to Poisson approximations of rare events in renewal processes

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

John L. Spouge
John L. Spouge*
Affiliation:
National Library of Medicine, USA
*
∗ Postal address: National Center for Biotechnology Information, National Library of Medicine, Bethesda, MD 20894, USA. Email address: spouge@nih.gov
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Abstract

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

Keywords

renewalsChen-Stein methodgenerating functions

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 554 - 569
Copyright
Copyright © Applied Probability Trust 2001 

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