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

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• 作者：Michael Drmota ; Alex Iksanov ; Martin Moehle ; Uwe Roesler
• 关键词：Asymptotic expansion ; Bolthausen– ; Sznitman coalescent ; Generating functions ; Random recursive trees ; Stable limit
• 刊名：Stochastic Processes and their Applications
• 出版年：2007
• 期刊代码：57_03044149
• 类别：mt
• 出版时间：October 2007
• 卷：117
• 期：10
• 页码：1404-1421
• 文件大小：329 K

摘要

We study the total branch length L_n of the Bolthausen–Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of L_n are presented. It is shown that converges to 1 in probability and that L_n, 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 L_n is closely related to X_n, 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.