{"title":"Scaling limits for the block counting process and the fixation line for a class of Λ-coalescents","authors":"M. Möhle, Benedict Vetter","doi":"10.30757/alea.v19-25","DOIUrl":null,"url":null,"abstract":". We provide scaling limits for the block counting process and the fixation line of Λ coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0 , 1] satisfies (cid:82) [0 , 1] u − 1 | Λ − bλ | (d u ) < ∞ for some b ≥ 0 . Here λ denotes the Lebesgue measure on [0 , 1] . The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein–Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0 . We split the generators into two parts by additively decomposing Λ into a ‘Bolthausen–Sznitman part’ bλ and a ‘dust part’ Λ − bλ and then prove the uniform convergence of both parts separately.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/alea.v19-25","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 2
Abstract
. We provide scaling limits for the block counting process and the fixation line of Λ coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0 , 1] satisfies (cid:82) [0 , 1] u − 1 | Λ − bλ | (d u ) < ∞ for some b ≥ 0 . Here λ denotes the Lebesgue measure on [0 , 1] . The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein–Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0 . We split the generators into two parts by additively decomposing Λ into a ‘Bolthausen–Sznitman part’ bλ and a ‘dust part’ Λ − bλ and then prove the uniform convergence of both parts separately.
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.