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标题：The Mittag--Leffler process and a scaling limit for the block counting process of the Bolthausen--Sznitman coalescent
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作者：Martin Möhle
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2015
卷号：XII
页码：35-53
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：The Mittag{Leer process X = (Xt)t0 is introduced. This Markovprocess has the property that its marginal random variables Xt are Mittag{Leerdistributed with parameter e􀀀t, t 2 [0;1), and the semigroup (Tt)t0 of X satises Ttf(x) = E(f(xe􀀀tXt)) for all x 0 and all bounded measurable functionsf : [0;1) ! R. Further characteristics of the process X are derived, for examplean explicit formula for the joint moments of its nite-dimensional distributions.The Mittag{Leer process turns out to be Siegmund dual to Neveu's continuousstatebranching process. The main result states that the block counting process ofthe Bolthausen{Sznitman n-coalescent, properly scaled, converges in the Skorohodtopology to the Mittag{Leer process X as the sample size n tends to innity. Weprovide an equivalent version of this convergence result involving stable distributions.
关键词：Block counting process; Bolthausen{Sznitman coalescent; continuous-;state branching process; marginal distributions; Mittag{Leer process; stable distribution; weak;convergence.