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Rarity and exponentiality: an extension of Keilson's theorem, with applications

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Rudolf Grübel  and
Marcus Reich
Rudolf Grübel*
Affiliation:
Universität Hannover
Marcus Reich*
Affiliation:
Universität Hannover
*
∗Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany.
∗Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany.
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Abstract

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We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.

Keywords

Exponential distributionfirst entrance timeLevenshtein distancelimit distributionOrnstein-Uhlenbeck processpattern matchingrandom stringregenerative process

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 393 - 406
Copyright
© Applied Probability Trust 2005 

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