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October 2019 Tree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent
Christina S. Diehl, Götz Kersting
Ann. Appl. Probab. 29(5): 2700-2743 (October 2019). DOI: 10.1214/19-AAP1462

Abstract

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.

Citation

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Christina S. Diehl. Götz Kersting. "Tree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent." Ann. Appl. Probab. 29 (5) 2700 - 2743, October 2019. https://doi.org/10.1214/19-AAP1462

Information

Received: 1 April 2018; Revised: 1 October 2018; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155057
MathSciNet: MR4019873
Digital Object Identifier: 10.1214/19-AAP1462

Subjects:
Primary: 60J75
Secondary: 60F05, 60J27, 92D25

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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