Electronic Communications in Probability

A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent

Martin Möhle and Helmut Pitters

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A spectral decomposition for the generator and the transition probabilities of the block counting process of the Bolthausen-Sznitman coalescent is derived. This decomposition is closely related to the Stirling numbers of the first and second kind. The proof is based on generating functions and exploits a certain factorization property of the Bolthausen-Sznitman coalescent. As an application we derive a formula for the hitting probability $h(i,j)$ that the block counting process of the Bolthausen-Sznitman coalescent ever visits state $j$ when started from state $i\ge j$. Moreover, explicit formulas are derived for the moments and the distribution function of the absorption time $\tau_n$ of the Bolthausen-Sznitman coalescent started in a partition with $n$ blocks. We provide an elementary proof for the well known convergence of $\tau_n-\log\log n$ in distribution to the standard Gumbel distribution. It is shown that the speed of this convergence is of order $1/\log n$.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 47, 11 pp.

Accepted: 23 July 2014
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60C05: Combinatorial probability 05C05: Trees 92D15: Problems related to evolution

absorption time Bolthausen-Sznitman coalescent Green matrix hitting probabilities spectral decomposition Stirling numbers

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Möhle, Martin; Pitters, Helmut. A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent. Electron. Commun. Probab. 19 (2014), paper no. 47, 11 pp. doi:10.1214/ECP.v19-3464. https://projecteuclid.org/euclid.ecp/1465316749

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