THE COLLISION SPECTRUM OF Λ-COALESCENTS

• 成果类型: Article
• 署名作者: Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander; Moehle, Martin
• 署名单位: University of London; Queen Mary University London; Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv; Eberhard Karls University of Tubingen
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1409
• 发表日期: 2018
• 页码: 3857-3883
• 关键词: bolthausen-sznitman coalescent; site-frequency-spectrum; random recursive trees; multiple collisions; beta-coalescents; regenerative compositions; asymptotic laws; SCALING LIMITS; markov-chains; number

摘要

Lambda-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially n singletons, we study the collision spectrum (X-n,X- k : 2 <= k <= n), where X-n,X- k counts, throughout the history of the process, the number of collisions involving exactly k blocks. Our focus is on the large n asymptotics of the joint distribution of the X-n,X- k's, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similar to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure Lambda in the vicinity of 0. In particular, for beta(a, b)-coalescents different types of limit distributions occur depending on whether 0 < a <= 1, 1 < a < 2, a = 2 or a > 2.