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, ∞)→(0, ∞) such that N_t/v(t)→1, almost surely, and in L^p for any p≥1, as t→0. Our approach relies on a novel martingale technique.