Scaling limits for a class of regular $\Xi$-coalescents

Abstract

The block counting process with initial state $n$ counts the number of blocks of an exchangeable coalescent ($\Xi$-coalescent) restricted to a sample of size $n$. This work provides scaling limits for the block counting process of regular $\Xi$-coalescents that stay infinite, including $\Xi$-coalescents with dust and a large class of dust-free $\Xi$-coalescents. The main convergence result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The existence of such a scaling depends on a sort of curvature condition of a particular function well-known from the literature. This curvature condition is intrinsically related to the behavior of the measure $\Xi$ 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.