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

By Jason Schweinsberg. 2000

109 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c5ba1e4ffa150ab477766ec374e880848d2bd2ce

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Alpha-Stable Branching and Beta-Coalescents (2005, methodology): “Remark 1.3. We say a �-coalescent (�t)t�0 comes down from infinity if �t almost surely has only a finite number of blocks for all t > 0. When � is the Beta(2 �,�) distribution, it is shown in [ 31 ] that the �-coalescent comes down from infinity if 1 < � < 2 but not when 0 < � � 1. Combining this ob”
Stochastic flows associated to coalescent processes (2003, background): “Necessary and sufficient conditions for the proper case to hold can be found in Schweinsberg [18]; see also Theorem 8 in Pitman [15] for the case of Λ-coalescents.”
Stochastic flows associated to coalescent processes. III. Limit theorems (2005, background): “To conclude this section, let us observe that Assumption (E) is closely related to the property for a Λ-coalescent to come down from infinity (cf Pitman [16] and Schweinsberg [18]).”
BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES (2006, background): “The reason we restrict our attention to 1 50 ] criterion).”
Combinatorial Stochastic Processes (2006, background): “(Coming down from infinity) [357, 395] Let (Π∞(t), t ≥ 0) be a Λ-coalescent starting with all singletons.”
Coalescents with Simultaneous Multiple Collisions (2000, methodology): “We record one simple lemma regarding the γb, which is proved in [19] for the Λ-coalescent by a direct calculation.”
A Classification of Coalescent Processes for Haploid Exchangeable Population Models (2001, unknown): “Pitman (1999) and Schweinsberg (2000a) study directly the class of coalescent processes with generator (9) without having a limit from a discrete population model in the background.”
Small-time behavior of beta coalescents (2006, unknown): “infinite for all t > 0. It is known (see Example 15 of [ 32 ]) that the Beta(2 − �,�) coalescent comes down from infinity if 1 32 ], but we give the short proof for completeness.”
On sampling distributions for coalescent processes with simultaneous multiple collisions (2006, background): “Basically, there are two approaches in the literature to coalescent processes with simultaneous multiple collisions, the ‘consistent rates approach’ of Pitman (1999) and Schweinsberg (2000b), and the ‘weak limit approach’ of Möhle and Sagitov (2001).”