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 Nt of blocks at any positive time t>0). We exhibit a deterministic function v:(0, ∞)→(0, ∞) such that Nt/v(t)→1, almost surely, and in Lp for any p≥1, as t→0. Our approach relies on a novel martingale technique.
"The Λ-coalescent speed of coming down from infinity." Ann. Probab. 38 (1) 207 - 233, January 2010. https://doi.org/10.1214/09-AOP475