On the Time Back to the Most Recent Common Ancestor and the External Branch Length of the Bolthausen - Sznitman Coalescent

F. Freund, M. Moehle

2009, v.15, Issue 3, 387-416

ABSTRACT

We study three particular functionals of the Bolthausen - Sznitman $n$-coalescent, namely the time $\tau_n$ back to the most recent common ancestor, the length $Z_n$ of an external branch chosen at random from the $n$ tips of the coalescent tree, and the number $C_n$ of collision events that occur before a randomly selected external branch coalesces with its closest branch. Explicit expressions and asymptotic expansions for all the moments of $\tau_n$ and $Z_n$ are provided. As a consequence, $\tau_n/\log\log n$ converges to $1$ almost surely and $\tau_n-\log\log n$ is asymptotically standard Gumbel distributed, a result which was first obtained by Goldschmidt and Martin [Electron. J. Probab. 10, 718-745]. Furthermore, we show that $Z_n\log n$ is asymptotically standard exponentially distributed and that $n^{-1}(\log n)C_n$ is asymptotically uniformly distributed. All weak convergence results are obtained via the method of moments. The proofs are mainly based on a singularity analysis of associated generating functions and are hence different from the probabilistic methods used by Goldschmidt and Martin.

Keywords: asymptotic expansions,Bolthausen - Sznitman coalescent,Gumbel distribution,generating functions,external branch,singularity analysis,time back to the most recent common ancestor

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