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2006 Stochastic flows associated to coalescent processes. III. Limit theorems
Jean Bertoin, Jean-Francois Le Gall
Illinois J. Math. 50(1-4): 147-181 (2006). DOI: 10.1215/ijm/1258059473

Abstract

We prove several limit theorems that relate coalescent processes to continuous-state branching processes. Some of these theorems are stated in terms of the so-called generalized Fleming-Viot processes, which describe the evolution of a population with fixed size, and are duals to the coalescents with multiple collisions studied by Pitman and others. We first discuss asymptotics when the initial size of the population tends to infinity. In that setting, under appropriate hypotheses, we show that a rescaled version of the generalized Fleming-Viot process converges weakly to a continuous-state branching process. As a corollary, we get a hydrodynamic limit for certain sequences of coalescents with multiple collisions: Under an appropriate scaling, the empirical measure associated with sizes of the blocks converges to a (deterministic) limit which solves a generalized form of Smoluchowski's coagulation equation. We also study the behavior in small time of a fixed coalescent with multiple collisions, under a regular variation assumption on the tail of the measure $\nu$ governing the coalescence events. Precisely, we prove that the number of blocks with size less than $\varepsilon x$ at time $(\varepsilon\nu([\varepsilon,1]))^{-1}$ behaves like $\varepsilon^{-1}\lambda_1(]0,x[)$ as $\varepsilon\to 0$, where $\lambda_1$ is the distribution of the size of one cluster at time $1$ in a continuous-state branching process with stable branching mechanism. This generalizes a classical result for the Kingman coalescent.

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Jean Bertoin. Jean-Francois Le Gall. "Stochastic flows associated to coalescent processes. III. Limit theorems." Illinois J. Math. 50 (1-4) 147 - 181, 2006. https://doi.org/10.1215/ijm/1258059473

Information

Published: 2006
First available in Project Euclid: 12 November 2009

zbMATH: 1110.60026
MathSciNet: MR2247827
Digital Object Identifier: 10.1215/ijm/1258059473

Subjects:
Primary: 60G09
Secondary: 60J25, 92D30

Rights: Copyright © 2006 University of Illinois at Urbana-Champaign

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Vol.50 • No. 1-4 • 2006
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