Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent

Michael Drmota, Alex Iksanov, Martin Moehle and Uwe Roesler

Stochastic Processes and their Applications, 2007, vol. 117, issue 10, 1404-1421

Abstract: We study the total branch length Ln of the Bolthausen-Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are applied to derive a corresponding limiting law for the total number of mutations for the Bolthausen-Sznitman coalescent with mutation rate r>0. Moreover, the results show that, for the Bolthausen-Sznitman coalescent, the total branch length Ln is closely related to Xn, the number of collision events that take place until there is just a single block. The proofs are mainly based on an analysis of random recursive equations using associated generating functions.

Keywords: Asymptotic; expansion; Bolthausen-Sznitman; coalescent; Generating; functions; Random; recursive; trees; Stable; limit (search for similar items in EconPapers)
Date: 2007
References: View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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