{"title":"金曼花的吸收时间、树长与甘贝尔分布","authors":"M. Möhle","doi":"10.15496/PUBLIKATION-9137","DOIUrl":null,"url":null,"abstract":"Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T are linear combinations of the values ζ(2m), m ∈ {0, . . . , bj/2c}, of the Riemann zeta function ζ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length Ln of the Kingman coalescent restricted to a sample of size n is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution\",\"authors\":\"M. Möhle\",\"doi\":\"10.15496/PUBLIKATION-9137\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T are linear combinations of the values ζ(2m), m ∈ {0, . . . , bj/2c}, of the Riemann zeta function ζ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length Ln of the Kingman coalescent restricted to a sample of size n is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.\",\"PeriodicalId\":48890,\"journal\":{\"name\":\"Markov Processes and Related Fields\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2015-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Markov Processes and Related Fields\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.15496/PUBLIKATION-9137\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Markov Processes and Related Fields","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.15496/PUBLIKATION-9137","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution
Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T are linear combinations of the values ζ(2m), m ∈ {0, . . . , bj/2c}, of the Riemann zeta function ζ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length Ln of the Kingman coalescent restricted to a sample of size n is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.
期刊介绍:
Markov Processes And Related Fields
The Journal focuses on mathematical modelling of today''s enormous wealth of problems from modern technology, like artificial intelligence, large scale networks, data bases, parallel simulation, computer architectures, etc.
Research papers, reviews, tutorial papers and additionally short explanations of new applied fields and new mathematical problems in the above fields are welcome.