The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent

TL;DR

This paper introduces the Mittag-Leffler process with specific distributional properties and demonstrates that the scaled block counting process of the Bolthausen-Sznitman coalescent converges to this process as the sample size grows.

Contribution

It establishes a new Markov process with Mittag-Leffler marginals and proves a scaling limit for the coalescent's block counting process.

Findings

The Mittag-Leffler process has explicit joint moments.

The block counting process converges to the Mittag-Leffler process.

The convergence is in the Skorohod topology.

Abstract

The Mittag-Leffler process $X=(X_t)_{t\ge 0}$ 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 $(T_t)_{t\ge 0}$ of $X$ satisfies $T_tf(x)={\mathbb E}(f(x^{e^{-t}}X_t))$ for all $x\ge 0$ and all bounded measurable functions $f:[0,\infty)\to{\mathbb R}$. Further characteristics of the process $X$ are derived, for example an explicit formula for the joint moments of its finite dimensional distributions. The main result states that the block counting process of the Bolthausen-Sznitman $n$-coalescent, properly scaled, converges in the Skorohod topology to the Mittag-Leffler process $X$ as the sample size $n$ tends to infinity.