TREE LENGTHS FOR GENERAL Λ-COALESCENTS AND THE ASYMPTOTIC SITE FREQUENCY SPECTRUM AROUND THE BOLTHAUSEN-SZNITMAN COALESCENT

• 成果类型: Article
• 署名作者: Diehl, Christina S.; Kersting, Goetz
• 署名单位: Goethe University Frankfurt
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1462
• 发表日期: 2019
• 页码: 2700-2743
• 关键词: branch length; number

摘要

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.