The joint fluctuations of the lengths of the Beta$(2-\alpha ,\alpha )$-coalescents

Abstract

We consider Beta$(2-\alpha ,\alpha )$-coalescents with parameter range $1<\alpha <2$ starting from $n$ leaves. The length ${\ell }_r^{(n)}$ of order $r$ in the $n$-Beta$(2-\alpha ,\alpha )$-coalescent tree is defined as the sum of the lengths of all branches that carry a subtree with $r$ leaves. We show that for any $s\in \mathbb{N}$ the vector of suitably centered and rescaled lengths of orders $1\le r\le s$ converges in distribution to a multivariate stable distribution as the number of leaves tends to infinity.

37 pages