On the speed of coming down from infinity for $\X$-coalescent processes

TL;DR

This paper investigates the speed at which certain $ ext{X}$-coalescent processes, which model genealogies in population genetics, decrease from infinitely many blocks to finitely many, providing a deterministic asymptotic description.

Contribution

It introduces a deterministic speed function for the small-time behavior of $ ext{X}$-coalescents that come down from infinity, extending previous techniques.

Findings

Established a deterministic speed function for $ ext{X}$-coalescents

Applied the method to a broad class of $ ext{X}$-coalescents

Provided almost sure small time asymptotics for the number of blocks

Abstract

The $\X$-coalescent processes were initially studied by M\"ohle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings' models. The $\X$-coalescents generalize $\Lambda$-coalescents, where now simultaneous multiple collisions of blocks are possible. The standard version starts with infinitely many blocks at time 0, and it is said to come down from infinity if its number of blocks becomes immediately finite, almost surely. This work builds on the technique introduced recently by Berstycki, Berestycki and Limic (2009), and exhibits a deterministic "speed" function -- an almost sure small time asymptotic to the number of blocks process, for a large class of $\X$-coalescents that come down from infinity.