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

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1.	Title	Title of document	A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent
2.	Creator	Author's name, affiliation, country	Martin Möhle; University of Tübingen, Germany; Germany
2.	Creator	Author's name, affiliation, country	Helmut Pitters; University of Oxford, UK; United Kingdom
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	absorption time; Bolthausen-Sznitman coalescent; Green matrix; hitting probabilities; spectral decomposition; Stirling numbers
3.	Subject	Subject classification	60J27; 60C05; 05C05; 92D15
4.	Description	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 $1/\log n$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2014-07-23
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/3464
10.	Identifier	Digital Object Identifier	10.1214/ECP.v19-3464
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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