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Möhle, Martin

The rate of convergence of the block counting process of exchangeable coalescents with dust. (English) Zbl 1477.60050

ALEA, Lat. Am. J. Probab. Math. Stat. 18, No. 2, 1195-1220 (2021).
Exchangeable coalescents (\(\Xi\)-coalescents) are continuous-time Markovian processes \(\prod= (\prod_t){t\geq 0}\) with values in the space \(\mathit{P}\) of partitions of \(N := \{1, 2,\cdots\}.\) Exchangeable coalescents with dust are studied. The rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups; they involve a particular concentration inequality arising in the context of Karlin’s infinite urn model. The rate of convergence is calculated for several examples of coalescents.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)

Cited in 3 Documents

MSC:

60F05 Central limit and other weak theorems
60J27 Continuous-time Markov processes on discrete state spaces
60J90 Coalescent processes
60G09 Exchangeability for stochastic processes
92D15 Problems related to evolution
47D07 Markov semigroups and applications to diffusion processes

Keywords:

Bernstein function; block counting process; concentration inequality; dust; exchangeable coalescent; rate of convergence; subordinator; continuous-time Markovian processes
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\textit{M. Möhle}, ALEA, Lat. Am. J. Probab. Math. Stat. 18, No. 2, 1195--1220 (2021; Zbl 1477.60050)
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References:

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