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2014 A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent
Martin Möhle, Helmut Pitters
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Electron. Commun. Probab. 19: 1-11 (2014). DOI: 10.1214/ECP.v19-3464

Abstract

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 $1/\log n$.

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Martin Möhle. Helmut Pitters. "A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent." Electron. Commun. Probab. 19 1 - 11, 2014. https://doi.org/10.1214/ECP.v19-3464

Information

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

zbMATH: 1334.60157
MathSciNet: MR3246966
Digital Object Identifier: 10.1214/ECP.v19-3464

Subjects:
Primary: 60J27
Secondary: 05C05, 60C05, 92D15

JOURNAL ARTICLE
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