On asymptotics of exchangeable coalescents with multiple collisions. (English) Zbl 1159.60016

Gnedin, Alexander; Iksanov, Alex; Möhle, Martin

J. Appl. Probab. 45, No. 4, 1186-1195 (2008).

This paper studies an exchangeable coalescent \(\prod_n=(\prod_n(t),\; t\geq0)\) with multiple collisions (\(\lambda\) – coalescent). This is a càdlàg (right continuous with left limits) Markov process that starts at \(t=0\) with \(n\) particles and is driven by rates determined by a finite characteristic measure \(\nu(dx)=x^{-2}\lambda(dx)\). The main focus of this note is on weak asymptotic behaviour of the number of collisions \(x_n\) and the absorption time \(\tau_n=\inf\{t\geq0:\prod_n(t)=1\}\), as \(n\) tends to \(\infty\). The principal result is criterion describing all possible limit lows of properly normalised and centred \(X_n\), include normal, stable with index \(1\leq\alpha\leq2\) and Mittag–Leffler distributions. Another result provides sufficient conditions under which \(\tau_n\) possesses the same collection of limit laws.

Reviewer: Nicko G. Gamkrelidze (Moskva)

MSC:

60G09	Exchangeability for stochastic processes
60C05	Combinatorial probability
60K05	Renewal theory

Keywords:

Absorption time; coupling; exchangeable coalescent; external branch length; multiple collisions; number of collisions; regenerative composition

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