The $Λ$-coalescent speed of coming down from infinity

Abstract

Consider a $Λ$-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.

Published in at http://dx.doi.org/10.1214/09-AOP475 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)