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Freund, Fabian; Möhle, Martin

On the size of the block of 1 for \(\varXi\)-coalescents with dust. (English) Zbl 1382.60109

Mod. Stoch., Theory Appl. 4, No. 4, 407-425 (2017).
Summary: We study the frequency process \(f_{1}\) of the block of 1 for a \(\varXi\)-coalescent \(\varPi\) with dust. If \(\varPi\) stays infinite, \(f_{1}\) is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-\(\varLambda\)-coalescents with \(\varLambda =\delta_{p}\), \(p\in [\frac{1}{2},1)\), \(f_{1}\) is not Markovian, whereas its jump chain is Markovian. For simple \(\varLambda\)-coalescents the distribution of \(f_{1}\) at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.

Cited in 2 Documents

MSC:

60J75 Jump processes (MSC2010)
60F15 Strong limit theorems
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G09 Exchangeability for stochastic processes
60J25 Continuous-time Markov processes on general state spaces
60F05 Central limit and other weak theorems

Keywords:

\(\varXi\)-coalescent; coalescent with dust; Poisson point process; minimal clade; exchangeability
PDFBibTeX XMLCite
\textit{F. Freund} and \textit{M. Möhle}, Mod. Stoch., Theory Appl. 4, No. 4, 407--425 (2017; Zbl 1382.60109)
Full Text: DOI arXiv

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