Stochastic flows associated to coalescent processes. III. Limit theorems

By Jean Bertoin, Jean-François Le Gall. 2005

137 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d4f4fa896e96bb35468f3facac060d7e86373698

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Citations in the corpus

The Λ-coalescent speed of coming down from infinity (2008, background): “Recently, Bertoin and Le Gall [6] observed that this condition is equivalent to the following requirement: define ψΛ(q) ≡ ψ(q) := ∫”
Proof(s) of the Lamperti representation of Continuous-State Branching Processes (2008, methodology): “This stochastic equation is similar to the ones shown to be satisﬁed by aﬃne processes in [9, (5.1)], [4] and [5, (9)].”
Small-time behavior of beta coalescents (2006, unknown): “It is not difficult to calculate (see Lemma 4 in [8]) that”
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees) (2009, background): “Such coalescent processes have since been the subject of many papers (see, for example, [3,5,6,21,23]).”
A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks (2008, methodology): “This construction and the duality has been extended to the so-called �-Fleming-Viot process, dual to the �-coalescents, and investigated by several authors, see, e.g., [DK99, BBC05, BLG03, BLG05, BLG06 ], or [BB07] for an overview.”
Strong solutions of jump-type stochastic equations (2012, unknown): “, [5, 6, 7, 14].”