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Möhle, Martin

A spectral decomposition for the block counting process and the fixation line of the beta(3,1)-coalescent. (English) Zbl 1406.60104

Electron. Commun. Probab. 23, Paper No. 102, 15 p. (2018).
Summary: A spectral decomposition for the generator of the block counting process of the \(\beta (3,1)\)-coalescent is provided. This decomposition is strongly related to Riordan matrices and particular Fuss-Catalan numbers. The result is applied to obtain formulas for the distribution function and the moments of the absorption time of the \(\beta (3,1)\)-coalescent restricted to a sample of size \(n\). We also provide the analog spectral decomposition for the fixation line of the \(\beta (3,1)\)-coalescent. The main tools in the proofs are generating functions and Siegmund duality. Generalizations to the \(\beta (a,1)\)-coalescent with parameter \(a\in (0,\infty )\) are discussed leading to fractional differential or integral equations.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J27 Continuous-time Markov processes on discrete state spaces
05C05 Trees
92D15 Problems related to evolution

Keywords:

absorption time; beta coalescent; fractional differential equation; Fuss-Catalan numbers; generating function; Riordan matrix; siegmund duality; spectral decomposition
PDFBibTeX XMLCite
\textit{M. Möhle}, Electron. Commun. Probab. 23, Paper No. 102, 15 p. (2018; Zbl 1406.60104)
Full Text: DOI Euclid

References:

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