The joint fluctuations of the lengths of the Beta(2−α,α)-coalescents

Abstract

We consider $Beta(2-\mathit{\alpha },\mathit{\alpha })$-coalescents with parameter range $1<\mathit{\alpha }<2$ starting from n leaves. The length ${\ell }_{\mathit{r}}^{(\mathit{n})}$ of order r in the n-$Beta(2-\mathit{\alpha },\mathit{\alpha })$-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 $\mathit{s}\in \mathbb{N}$ the vector of suitably centered and rescaled lengths of orders $1\le \mathit{r}\le \mathit{s}$ converges in distribution to a multivariate stable distribution as the number of leaves tends to infinity.