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Iksanov, Alex; Marynych, Alex; Möhle, Martin

On the number of collisions in beta(\(2, b\))-coalescents. (English) Zbl 1208.60081

Bernoulli 15, No. 3, 829-845 (2009).
Coalescent processes are stochastic Markov processes of partitions on the natural numbers (for example the Kingman coalescence or the Bolthausen-Sznitman coalescence). Recently J. Pitman [Ann. Probab. 27, No. 4, 1870–1902 (1999; Zbl 0963.60079)] and S. Sagitov [J. Appl. Probab. 36, No.4, 1116–1125 (1999; Zbl 0962.92026)] introduced a general concept of (so-called) \(\Lambda\)-coalescents, where \(\Lambda\) denotes a finite measure on \([0,1]\).
The present paper deals with the special case of \(\beta(2,b)\)-coalescents, where \(\Lambda = \beta(2,b)\) is the beta distribution. In particular the authors derive asymptotic expansions for the moments of the number of collisions \(X_n\), and they prove a strong law of large numbers and a central limit theorem.
These results complement previous results for \(\beta(a,b)\)-coalescents with \(a\neq 2\). The case \(a=2\) seems to be a kind of borderline situation since it (seems that it) requires a different proof technique. Actually the authors apply the so-called contraction method to the stochastic recurrence \(X_n = X_{n-I_n} + 1\), where the (discrete) distribution of \(I_n\) is given in terms of \(\Lambda=\beta(2,b)\).
Reviewer: Michael Drmota (Wien)

Cited in 10 Documents

MSC:

60J75 Jump processes (MSC2010)
60F05 Central limit and other weak theorems
92D15 Problems related to evolution

Keywords:

asymptotics of moments; beta-coalescent; number of collisions; random regenerative composition; recursion with random indices

Citations:

Zbl 0963.60079; Zbl 0962.92026
PDFBibTeX XMLCite
\textit{A. Iksanov} et al., Bernoulli 15, No. 3, 829--845 (2009; Zbl 1208.60081)
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References:

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