A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent. (English) Zbl 1334.60157

Möhle, Martin; Pitters, Helmut

Electron. Commun. Probab. 19, Paper No. 47, 11 p. (2014).

Summary: 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 \(i\geq j\). 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 \(1/\log n\).

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60C05	Combinatorial probability
05C05	Trees

Keywords:

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