The $\Lambda$-coalescent speed of coming down from infinity

TL;DR

This paper introduces a deterministic function describing the rate at which a $ ext{Lambda}$-coalescent process transitions from infinitely many blocks to finitely many, using a new martingale approach.

Contribution

It provides a novel martingale-based method to characterize the speed of coming down from infinity for $ ext{Lambda}$-coalescents.

Findings

Established a deterministic function $v(t)$ for the process speed.

Proved $N_t/v(t) o 1$ almost surely and in $L^p$ as $t o 0$.

Introduced a new martingale technique for analyzing coalescent processes.

Abstract

Consider a $\Lambda$-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit a deterministic function $v:(0,\infty)\to(0,\infty)$ such that $N_t/v(t)\to1$, almost surely, and in $L^p$ for any $p\geq1$, as $t\to0$. Our approach relies on a novel martingale technique.