## Electronic Communications in Probability

### A Necessary and Sufficient Condition for the $\Lambda$-Coalescent to Come Down from Infinity.

Jason Schweinsberg

#### Abstract

Let $\Pi_{\infty}$ be the standard $\Lambda$-coalescent of Pitman, which is defined so that $\Pi_{\infty}(0)$ is the partition of the positive integers into singletons, and, if $\Pi_n$ denotes the restriction of $\Pi_{\infty}$ to $\{ 1,\ldots, n \}$, then whenever $\Pi_n(t)$ has $b$ blocks, each $k$-tuple of blocks is merging to form a single block at the rate $\lambda_{b,k}$, where $\lambda_{b,k} = \int_0^1 x^{k-2} (1-x)^{b-k} \Lambda(dx)$ for some finite measure $\Lambda$. We give a necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity'', which means that the partition $\Pi_{\infty}(t)$ almost surely consists of only finitely many blocks for all $t \gt 0$. We then show how this result applies to some particular families of $\Lambda$-coalescents.

#### Article information

Source
Electron. Commun. Probab., Volume 5 (2000), paper no. 1, 1-11.

Dates
Accepted: 23 November 1999
First available in Project Euclid: 2 March 2016

https://projecteuclid.org/euclid.ecp/1456943494

Digital Object Identifier
doi:10.1214/ECP.v5-1013

Mathematical Reviews number (MathSciNet)
MR1736720

Zentralblatt MATH identifier
0953.60072

Subjects
Primary: 60J75: Jump processes
Secondary: 60G09: Exchangeability

Keywords
coalescent Kochen-Stone Lemma

Rights
Schweinsberg, Jason. A Necessary and Sufficient Condition for the $\Lambda$-Coalescent to Come Down from Infinity. Electron. Commun. Probab. 5 (2000), paper no. 1, 1--11. doi:10.1214/ECP.v5-1013. https://projecteuclid.org/euclid.ecp/1456943494