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

Part of: Limit theorems Markov processes

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 

References

Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.Google Scholar
Aldous, D. and Fill, J. A. (2004). Reversible Markov chains and Random Walks on Graphs. In preparation. Available at http://www.stat.berkeley.edu/∼aldous.Google Scholar
Billingsley, P. (1968). Weak Convergence of Probability Measures. John Wiley, New York.Google Scholar
Billingsley, P. (1986). Probability and Measure, 2nd edn. John Wiley, New York.Google Scholar
Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.Google Scholar
Brémaud, P. (1999). Markov Chains. Gibbs Fields, Monte Carlo Simulation and Queues. Springer, New York.Google Scholar
Chung, K. L. (1967). Markov Chains With Stationary Transition Probabilities, 2nd edn. Springer, Berlin.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Gerber, H. U. and Li, S.-Y. R. (1981). The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch. Process. Appl. 11, 101–108.CrossRefGoogle Scholar
Guibas, L. J. and Odlyzko, A. M. (1981). String overlaps, pattern matching and nontransitive games. J. Combin. Theory A 30, 183–208.CrossRefGoogle Scholar
Gusfield, D. (1997). Algorithms on Strings, Trees, and Sequences. Cambridge University Press.Google Scholar
Kac, M. (1959). Probability and Related Topics in Physical Sciences (Proc. Summer Seminar, Boulder, CO), Vol. 1. Interscience, London.Google Scholar
Keilson, J. (1966). A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 866–870.CrossRefGoogle Scholar
Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality (Appl. Math. Sci. 28). Springer, New York.CrossRefGoogle Scholar
Kemperman, J. H. B. (1961). The Passage Problem for a Stationary Markov Chain (Statist. Res. Monogr.), Vol. 1. The University of Chicago Press.Google Scholar
Li, S.-Y. R. (1980). A martingale approach to the study of the occurrence of sequence patterns in repeated experiments. Ann. Prob. 8, 1171–1176.Google Scholar
Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.Google Scholar
Reich, M. (2004). Asymptotische Exponentialität und die Approximation von Wartezeitverteilungen für Pattern in zufälligen Zeichenketten. , Universität Hannover.Google Scholar
Rosenkrantz, W. A. and Dorea, C. C. Y. (1980). Limit theorems for Markov processes via a variant of the Trotter–Kato theorem. J. Appl. Prob. 17, 704–715.Google Scholar
Rudander, J. (1996). On the first occurrence of a given pattern in a semi-Markov process. , Uppsala University.Google Scholar
Waterman, M. S. (1995). Introduction to Computational Biology. Maps, Sequences and Genomes. Chapman and Hall, London.Google Scholar