On the number of collisions in $\Lambda$-coalescents

TL;DR

This paper investigates the total number of collisions in $\Lambda$-coalescent processes, demonstrating linear growth and stable limit laws under specific measure conditions near zero.

Contribution

It establishes new asymptotic results for collision counts in $\Lambda$-coalescents with measures exhibiting power-like behavior near zero.

Findings

Total collisions grow linearly with initial particles

Collision count follows a stable limit law

Results depend on measure's behavior near zero

Abstract

We examine the total number of collisions $C_n$ in the $\Lambda$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $\Lambda$ near 0 with exponent $0<\alpha<1$.