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

@article{Schweinsberg2000ANA,
title={A Necessary and Sufficient Condition for the \$\Lambda\$-Coalescent to Come Down from Infinity.},
author={Jason Schweinsberg},
journal={Electronic Communications in Probability},
year={2000},
volume={5},
pages={1-11}
}
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…
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