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：451-476
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：Representation of coalescent processes using pruning of trees has beenused by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and byAbraham and Delmas for the β(3/2, 1/2)-coalescent. By considering a pruning procedureon stable Galton-Watson tree with n labeled leaves, we give a representationof the discrete β(1+α, 1−α)-coalescent, with α ∈ [1/2, 1) starting from the trivialpartition of the n first integers. The construction can also be made directly on thestable continuum L´evy tree, with parameter 1/α, simultaneously for all n. Thisrepresentation allows to use results on the asymptotic number of coalescence eventsto get the asymptotic number of cuts in stable Galton-Watson tree (with infinitevariance for the offspring distribution) needed to isolate the root. Using convergenceof the stable Galton-Watson tree conditioned to have infinitely many leaves,one can get the asymptotic distribution of blocks in the last coalescence event inthe β(1 + α, 1 − α)-coalescent.
