The rate of convergence of the block counting process of exchangeable coalescents with dust

Abstract

Exchangeable coalescents with dust are studied. The rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups and involve a particular concentration inequality arising in the context of Karlin's infinite urn model. The rate of convergence is calculated for several examples of coalescents.

Scaling limits for the block counting process and the fixation line of a class of $Λ$-coalescents

“…If [0,1] u −1 Λ(du) < ∞, then the Λ-coalescent has dust and Assumption A is satisfied with b = 0. Corollary 3 below has been proven in [9] and [15]. In both articles the blocks of the coalescent are allowed to merge simultaneously.…”

“…In both articles the blocks of the coalescent are allowed to merge simultaneously. In [15] the convergence of the generators has been proven and even a rate of convergence has been determined. In this article the uniform convergence of the generators is going to be proven as well, but with different techniques.…”

“…with initial value X 0 = 0 has an unique (F t ) t≥0 -adapted solution X = (X t ) t≥0 with càdlàg paths. The solution of (15) or the corresponding semigroup are hence sometimes called Ornstein-Uhlenbeck type or generalized Ornstein-Uhlenbeck process or semigroup. It holds that…”

Scaling limits for a class of regular $Ξ$-coalescents

“…For the β(1, b)-coalescent scaling limits have already been obtained in [28], and we clarify beforehand the relation between [28] and this work. Example 2 studies the Λ-coalescent introduced in [26], where the measure Λ is a negative logarithmic gamma distribution (NLG-coalescent). Example 3 provides a simple dust-free Λ-coalescent which nevertheless satisfies κ = 0.…”

“…t |) t≥0 for large initial state, more precisely, to determine scaling functions v(n, t) for which N (n) t /v(n, t) converges in distribution as n → ∞. For coalescents with dust it is proven in [15] and [26] with different methods that (N (n) t /n) t≥0 converges in the space D [0,1] [0, ∞) of càdlàg paths endowed with the Skorohod topology to the so-called frequency of singletons process as n → ∞. The Bolthausen-Sznitman coalescent in which the driving measure Λ is the uniform distribution on [0, 1] has been thoroughly studied in the literature and is an example of a dust-free Λ-coalescent that stays infinite.…”