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A Necessary and Sufficient Condition for the Lambda-Coalescent to Come Down from Infinity.
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Jason Schweinsberg, University of California, Berkeley
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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.
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Full text: PDF
Pages: 1-11
Published on: November 23, 1999
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Electronic Communications in Probability. ISSN: 1083-589X |
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Context