The Annals of Applied Probability

Tree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent

Christina S. Diehl and Götz Kersting

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We study tree lengths in $\Lambda $-coalescents without a dust component from a sample of $n$ individuals. For the total length of all branches and the total length of all external branches, we present laws of large numbers in full generality. The other results treat regularly varying coalescents with exponent 1, which cover the Bolthausen–Sznitman coalescent. The theorems contain laws of large numbers for the total length of all internal branches and of internal branches of order $a$ (i.e., branches carrying $a$ individuals out of the sample). These results immediately transform to sampling formulas in the infinite sites model. In particular, we obtain the asymptotic site frequency spectrum of the Bolthausen–Sznitman coalescent. The proofs rely on a new technique to obtain laws of large numbers for certain functionals of decreasing Markov chains.

Article information

Ann. Appl. Probab., Volume 29, Number 5 (2019), 2700-2743.

Received: April 2018
Revised: October 2018
First available in Project Euclid: 18 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60J75: Jump processes
Secondary: 60F05: Central limit and other weak theorems 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)

$\Lambda $-coalescent Bolthausen–Sznitman coalescent law of large numbers tree length infinite sites model sampling formula site frequency spectrum decreasing Markov chain


Diehl, Christina S.; Kersting, Götz. Tree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent. Ann. Appl. Probab. 29 (2019), no. 5, 2700--2743. doi:10.1214/19-AAP1462.

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