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

作者：Moehle Martin; Pitters Helmut

来源：Electronic Communications in Probability, 2014, 19(0): 1-11.

DOI：10.1214/ECP.v19-3464

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

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 BolthausenSznitman coalescent ever visits state j when started from state i >= j. Moreover, explicit formulas are derived for the moments and the distribution function of the absorption time T-n of the Bolthausen-Sznitman coalescent started in a partition with n blocks. We provide an elementary proof for the well known convergence of T-n loglogn in distribution to the standard Gumbel distribution. It is shown that the speed of this convergence is of order 1/logn.

• 出版日期2014-7-23