## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 50, Number 1-4 (2006), 147-181.

### Stochastic flows associated to coalescent processes. III. Limit theorems

Jean Bertoin and Jean-Francois Le Gall

#### 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.

#### Article information

**Source**

Illinois J. Math. Volume 50, Number 1-4 (2006), 147-181.

**Dates**

First available in Project Euclid: 12 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258059473

**Mathematical Reviews number (MathSciNet)**

MR2247827

**Zentralblatt MATH identifier**

1110.60026

**Subjects**

Primary: 60G09: Exchangeability

Secondary: 60J25: Continuous-time Markov processes on general state spaces 92D30: Epidemiology

#### Citation

Bertoin, Jean; Le Gall, Jean-Francois. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006), no. 1-4, 147--181.https://projecteuclid.org/euclid.ijm/1258059473