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On \(\Lambda \)-coalescents with dust component. (English) Zbl 1242.60077

There is a class of \(\Lambda\)-coalescents \(\Pi_\infty\) which stay infinite because infinitely many of the original particles do not engage in collisions before any given time. For such coalescents restricted to \(n\) particles (denote them by \(\Pi_n\)), the present paper investigates the weak convergence of the number of collisions and the absorption time as \(n\) becomes large. The paper proves that most of the collisions of \(\Pi_n\) will involve some of the original \(n\) particles for large \(n\). In particular, the behavior of the number of collisions and the absorption time can be derived from that of analogous quantities associated with the evolution of singletons. The total frequency of singletons in \(\Pi_\infty\) evolves like a multiplicative subordinator. For \(\Pi_n\), the engagement of original \(n\) particles in their first collisions follows a simple Markovian process which has been studied in the context of regenerative composition structures derived from subordinators. A coupling of \(\Pi_\infty\) with a subordinator enables one to apply known results about the level-passage for subordinators, and about the asymptotics of regenerative composition structures.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
60G09 Exchangeability for stochastic processes
60K05 Renewal theory
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