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Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent

Part of: Markov processes Combinatorial probability Graph theory

Published online by Cambridge University Press:  30 March 2016

Martin Möhle
Martin Möhle*
Affiliation:
Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: [email protected].
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Abstract

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The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = limn→∞h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.

Keywords

Asymptotic hitting probabilityBolthausen-Sznitman coalescentgenerating function

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60C05: Combinatorial probability
Secondary: 05C05: Trees 92D15: Problems related to evolution
Type
Part 3. Biological applications
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 87 - 97
Copyright
Copyright © Applied Probability Trust 2014 

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