Scaling limits for a class of regular \(\Xi\)-coalescents. (English) Zbl 1518.60086

Möhle, Martin; Vetter, Benedict

Stochastic Processes Appl. 162, 387-422 (2023).

Summary: Let \(N_t^{(n)}\) denote the number of blocks in a \(\Xi\)-coalescent restricted to a sample of size \(n\in\mathbb{N}\) after time \(t\geq 0\). Under the assumption of a certain curvature condition on a function well-known from the literature, we prove the existence of sequences \((v(n,t))_{n\in\mathbb{N}}\) for which \((\log N_t^{(n)}-\log v(n,t))_{t\geq 0}\) converges to an Ornstein-Uhlenbeck type process as \(n\to\infty\). The curvature condition is intrinsically related to the behavior of \(\Xi\) 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.

MSC:

60J90	Coalescent processes
60J27	Continuous-time Markov processes on discrete state spaces

Keywords:

block counting process; fixation line; Ornstein-Uhlenbeck-type process; regular coalescent; simultaneous multiple collisions; weak convergence

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