The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent. (English) Zbl 1329.60271

Möhle, Martin

ALEA, Lat. Am. J. Probab. Math. Stat. 12, No. 1, 35-53 (2015).

Summary: The Mittag-Leffler process \(X = (X_t)_{t\geq 0}\) is introduced. This Markov process has the property that its marginal random variables \(X_t\) are Mittag-Leffler distributed with parameter \(e^{-t}, t \in [0, \infty)\), and the semigroup \((T_t)_{t\geq 0}\) of \(X\) satisfies \(T_tf (x) = \mathbb E(f (x^{e^{-t}}X_t))\) for all \(x \geq 0\) and all bounded measurable functions \(f : [0, \infty) \rightarrow R\). Further characteristics of the process \(X\) are derived, for example an explicit formula for the joint moments of its finite-dimensional distributions.
The Mittag-Leffler process turns out to be Siegmund dual to Neveu’s continuous-state branching process. The main result states that the block counting process of the Bolthausen-Sznitman \(n\)-coalescent, properly scaled, converges in the Skorokhod topology to the Mittag-Leffler process \(X\) as the sample size \(n\) tends to infinity. We provide an equivalent version of this convergence result involving stable distributions.

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60F17	Functional limit theorems; invariance principles
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F05	Central limit and other weak theorems
60E07	Infinitely divisible distributions; stable distributions
92D15	Problems related to evolution

Keywords:

Mittag-Leffler process; block counting process; Bolthausen-Sznitman coalescent; scaling limit; continuous-state branching process; marginal distributions; stable distributions; weak convergence