The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

• 成果类型: Article
• 署名作者: Blum, Michael G. B.; Francois, Olivier; Janson, Svante
• 署名单位: Uppsala University; Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000547
• 发表日期: 2006
• 页码: 2195-2214
• 关键词: central moments; wiener index; shape; quicksort; imbalance; models; diversification; yule

摘要

For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenctic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.