Scaling limits for a class of regular [formula omitted]-coalescents

Abstract

Let Nt(n) denote the number of blocks in a Ξ-coalescent restricted to a sample of size n∈N after time t≥0. Under the assumption of a certain curvature condition on a function well-known from the literature, we prove the existence of sequences (v(n,t))n∈N for which (logNt(n)−logv(n,t))t≥0 converges to an Ornstein–Uhlenbeck type process as n→∞. The curvature condition is intrinsically related to the behavior of Ξ near the origin. The method of proof is to show the uniform convergence of the associated generators. Via Siegmund duality an analogous result for the fixation line is proven. Several examples are studied.