The class of coalescent processes with simultaneous multiple collisions (Ξ-coalescents)
without proper frequencies is considered. We study the asymptotic behavior of the external …

We prove asymptotic normality of the total length of external branches in the Kingman
coalescent. The proof uses an embedded Markov chain, which can be described as follows …

We study scaling limits of non-increasing Markov chains with values in the set of non-
negative integers, under the assumption that the large jump events are rare and happen at …

In the Kingman coalescent tree the length of order r is defined as the sum of the lengths of all
branches that support r leaves. For r=1 these branches are external, while for r\ge2 they are …

In this paper we provide a systematic exposition of basic properties of integrated distribution
and quantile functions. We define these transforms in such a way that they characterize any …

In this paper, we consider Beta (2− α, α)(with 1< α< 2) and related Λ-coalescents. If T (n)
denotes the length of a randomly chosen external branch of the n-coalescent, we prove the …

Kingman's theory of partition structures relates, via a natural sampling procedure, finite
partitions to hypothetical infinite populations. Explicit formulas for distributions of such …

Λ-coalescents model the evolution of a coalescing system in which any number of
components randomly sampled from the whole may merge into larger blocks. This survey …

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The
class in focus covers, for instance, the beta (a, b)-coalescents with a> 1. We show that some …

A hierarchy on a set S, also called a total partition of S, is a collection HH of subsets of S
such that S ∈ HS∈ H, each singleton subset of S belongs to HH, and if A, B ∈ HA, B∈ H …