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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Regenerative partition structures Alexander Gnedin. | Regenerative partition structures Alexander Gnedin Utrecht University e-mail gnedin@math.uu.nl and Jim Pitman University of California Berkeley pitman@stat.Berkeley.EDU Submitted Aug 6 2004 Accepted Nov 4 2004 Published Jan 7 2005 Mathematics Subject Classifications 60G09 60C05. Keywords partition structure deletion kernel regenerative composition structure Abstract A partition structure is a sequence of probability distributions for Kn a random partition of n such that if Kn is regarded as a random allocation of n unlabeled balls into some random number of unlabeled boxes and given rn some x of the n balls are removed by uniform random deletion without replacement the remaining random partition of n x is distributed like rn-x for all 1 x n. We call a partition structure regenerative if for each n it is possible to delete a single box of balls from n in such a way that for each 1 x n given the deleted box contains x balls the remaining partition of n x balls is distributed like Kn-x. Examples are provided by the Ewens partition structures which Kingman characterised by regeneration with respect to deletion of the box containing a uniformly selected random ball. We associate each regenerative partition structure with a corresponding regenerative composition structure which as we showed in a previous paper is associated in turn with a regenerative random subset of the positive halfline. Such a regenerative random set is the closure of the range of a subordinator that is an increasing process with stationary independent increments . The probability distribution of a general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator for which exponent an integral representation is provided by the Levy-Khintchine formula. The extended Ewens family of partition structures previously studied by Pitman and Yor with two parameters a 0 is characterised for 0 a 1 and 0 0 by regeneration with respect to deletion of each