A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity.

September, 1999
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Jason Schweinsberg
ECP Vol 5 (2000) Paper 1

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 \begin{displaymath} \lambda_{b,k} = \int_0^1 x^{k-2} (1-x)^{b-k} \: \Lambda(dx) \end{displaymath} 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 $t > 0$. We then show how this result applies to some particular families of $\Lambda$-coalescents.

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