On the block counting process and the fixation line of the Bolthausen-Sznitman coalescent

TL;DR

This paper provides spectral decompositions and explicit formulas for the block counting process and fixation line of the Bolthausen-Sznitman coalescent, demonstrating their convergence to well-known stochastic processes as sample size grows.

Contribution

It introduces explicit spectral decompositions and convergence results for key processes of the Bolthausen-Sznitman coalescent, linking them to classical stochastic processes.

Findings

Spectral decompositions for generators and transition probabilities

Explicit formulas for hitting probabilities and absorption times

Convergence to Mittag-Leffler process and Neveu's CSBP as sample size increases

Abstract

The block counting process and the fixation line of the Bolthausen-Sznitman coalescent are analyzed. Spectral decompositions for their generators and transition probabilities are provided leading to explicit expressions for functionals such as hitting probabilities and absorption times. It is furthermore shown that the block counting process and the fixation line of the Bolthausen-Sznitman $n$-coalescent, properly scaled, converge in the Skorohod topology to the Mittag-Leffler process and Neveu's continuous-state branching process respectively as the sample size $n$ tends to infinity. Strong relations to Siegmund duality and to Mehler semigroups and self-decomposability are pointed out.