The $\Lambda$-coalescent speed of coming down from infinity
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 $N_t/v(t)\to 1$, almost surely and in $L^p$ for any $p\geq 1$, as $t\to 0$. Our approach relies on a novel martingale technique.