Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent

  title={Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent},
  author={Martin M{\"o}hle},
  journal={Stochastic Processes and their Applications},
  • M. Möhle
  • Published 1 November 2010
  • Mathematics
  • Stochastic Processes and their Applications
On the external branches of coalescents with multiple collisions
A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Lambda-coalescents) is provided and results show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen-Sznitman coalescent.
A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (-coalescents) is provided. This recursion is used to derive asymptotic expansions as the
Genealogies of regular exchangeable coalescents with applications to sampling
This article considers a model of genealogy corresponding to a regular exchangeable coalescent started from a large finite configuration, and undergoing neutral mutations, and derived analogous results for the number of active mutation-free lineages and the combined lineage lengths.
Scaling limits for a class of regular $\Xi$-coalescents
The block counting process with initial state n counts the number of blocks of an exchangeable coalescent (Ξ-coalescent) restricted to a sample of size n . This work provides scaling limits for the
The symmetric coalescent and Wright–Fisher models with bottlenecks
We define a new class of $\Xi$-coalescents characterized by a possibly infinite measure over the non negative integers. We call them symmetric coalescents since they are the unique family of
External branch lengths of Λ-coalescents without a dust component
Λ -coalescents model genealogies of samples of individuals from a large population by means of a family tree. The tree’s leaves represent the individuals, and the lengths of the adjacent edges
The total external length of the evolving Kingman coalescent
The evolving Kingman coalescent is the tree-valued process which records the time evolution undergone by the genealogies of Moran populations. We consider the associated process of total external
Asymptotic genealogies for a class of generalized Wright–Fisher models
We study a class of Cannings models with population size N having a mixed multinomial offspring distribution with random success probabilities W1, . . . ,WN induced by independent and identically
Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution
Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T


On Asymptotics of Exchangeable Coalescents with Multiple Collisions
We study the number of collisions, X n , of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite
Coalescents with multiple collisions
k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's
On sampling distributions for coalescent processes with simultaneous multiple collisions
Recursions for a class of sampling distributions of allele configurations are derived for the situation where the genealogy of the underlying population is modelled by a coalescent process with
On the number of segregating sites for populations with large family sizes
  • M. Möhle
  • Mathematics
    Advances in Applied Probability
  • 2006
We present recursions for the total number, S n , of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate
The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized by Mohle and Sagitov (2001) in terms of a sequence of measures defined on the
Asymptotic results on the length of coalescent trees
We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations
Coalescents with Simultaneous Multiple Collisions
We study a family of coalescent processes that undergo ``simultaneous multiple collisions,'' meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers
Self-similar scaling limits of non-increasing Markov chains
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 rates that behave
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with