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

   
@article{ECP1013,
	author = {Jason Schweinsberg},
	title = {A Necessary and Sufficient Condition for the Lambda-Coalescent to Come Down from Infinity.},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {5},
	year = {1999},
	keywords = {coalescent, Kochen-Stone Lemma},
	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 $t > 0$. We then show how this result applies to some particular families of $\Lambda$-coalescents.},
	pages = {no. 1, 1-11},
	issn = {1083-589X},
	doi = {10.1214/ECP.v5-1013},    
        url = {http://ecp.ejpecp.org/article/view/1013}}