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Stochastic flows associated to coalescent processes. III. Limit theorems


Bertoin, Jean; Le Gall, J-F (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois Journal of Mathematics, 50(1-4):147-181.

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 ν governing the coalescence events. Precisely, we prove that the number of blocks with size less than εx at time (εν([ε,1]))−1 behaves like ε−1λ1(]0,x[) as ε→0, where λ1 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.

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 ν governing the coalescence events. Precisely, we prove that the number of blocks with size less than εx at time (εν([ε,1]))−1 behaves like ε−1λ1(]0,x[) as ε→0, where λ1 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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2006
Deposited On:29 May 2013 14:26
Last Modified:05 Apr 2016 16:47
Publisher:University of Illinois Press
ISSN:0019-2082
Free access at:Official URL. An embargo period may apply.
Official URL:http://projecteuclid.org/euclid.ijm/1258059473
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1110.60026
http://www.ams.org/mathscinet-getitem?mr=2247827

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