THE INTERNAL BRANCH LENGTHS OF THE KINGMAN COALESCENT

• 成果类型: Article
• 署名作者: Dahmer, Iulia; Kersting, Goetz
• 署名单位: Goethe University Frankfurt
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1024
• 发表日期: 2015
• 页码: 1325-1348
• 关键词: Asymptotics

摘要

In the Kingman coalescent tree the length of order r is defined as the sum of the lengths of all branches that support r leaves. For r = 1 these branches are external, while for r >= 2 they are internal and carry a subtree with r leaves. In this paper we prove that for any s is an element of N the vector of resealed lengths of orders 1 <= r <= s converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument which shows that for any r >= 2 the (internal) length of order r behaves asymptotically in the same way as the length of order 1 (i.e., the external length).