@article{ECP3464,
author = {Martin Möhle and Helmut Pitters},
title = {A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {absorption time; Bolthausen-Sznitman coalescent; Green matrix; hitting probabilities; spectral decomposition; Stirling numbers},
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 $1/\log n$.},
pages = {no. 47, 1-11},
issn = {1083-589X},
doi = {10.1214/ECP.v19-3464},
url = {http://ecp.ejpecp.org/article/view/3464}}