The Annals of Applied Probability

The fixation line in the ${\Lambda}$-coalescent

Olivier Hénard

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Abstract

We define a Markov process in a forward population model with backward genealogy given by the $\Lambda$-coalescent. This Markov process, called the fixation line, is related to the block counting process through its hitting times. Two applications are discussed. The probability that the $n$-coalescent is deeper than the $(n-1)$-coalescent is studied. The distribution of the number of blocks in the last coalescence of the $n$-$\operatorname{Beta}(2-\alpha,\alpha)$-coalescent is proved to converge as $n\rightarrow\infty$, and the generating function of the limiting random variable is computed.

Article information

Source
Ann. Appl. Probab. Volume 25, Number 5 (2015), 3007-3032.

Dates
Received: August 2013
Revised: September 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1438261058

Digital Object Identifier
doi:10.1214/14-AAP1077

Mathematical Reviews number (MathSciNet)
MR3375893

Zentralblatt MATH identifier
1325.60124

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60G55: Point processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Coalescent Markov chain duality hitting times

Citation

Hénard, Olivier. The fixation line in the ${\Lambda}$-coalescent. Ann. Appl. Probab. 25 (2015), no. 5, 3007--3032. doi:10.1214/14-AAP1077. http://projecteuclid.org/euclid.aoap/1438261058.


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