On the time back to the most recent common ancestor and the external branch length of the Boltzhausen-Sznitman coalescent. (English) Zbl 1203.60110

Freund, F.; Möhle, M.

Markov Process. Relat. Fields 15, No. 3, 387-416 (2009).

The paper studies three important functionals of the Boltzhausen-Sznitman \(n\)-coalescent process (the particular case of \(\Lambda\)-coalescent processes, where \(\Lambda\) is the uniform measure on \([0,1]\)), namely the time \(\tau_n\) back to the most 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 branches.
Exact (cumbersome) and asymptotic expressions are given for the moments of \(\tau_n\) and \(Z_n\), as well as an asymptotic expression for the moments of \(C_n\). Then it is shown that \(\tau_n / \log \log n\rightarrow 1\) as \(n\rightarrow +\infty\) a.s. The previous result by C. Goldschmidt and J. B. Martin [Electron. J. Probab. 10, Paper No. 21, 718–745, electronic only (2005; Zbl 1109.60060)] that \(\tau_n - \log \log n\) converges in distribution to a standard Gumbel is here proven by analytical instead of probabilistic methods. It is also shown that \(Z_n \log n\) and \(C_n (\log n)/n\) converge in distribution, respectively, to a standard exponential and to a uniform (on \([0,1])\). Proofs are based mainly on a singularity analysis of associated generating functions.

Reviewer: Carlos A. Braumann (Évora)

MSC:

60J25	Continuous-time Markov processes on general state spaces
92D15	Problems related to evolution
60F05	Central limit and other weak theorems
60C05	Combinatorial probability
05C05	Trees
92D10	Genetics and epigenetics

Keywords:

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

Citations:

Zbl 1109.60060