期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2007
卷号：69
期号：02
出版社：Indian Statistical Institute
摘要：To fix the background and notations, we shall first briefly revisit some aspects of the following Ewens-like randomized occupancy problem: assume that distinguishable particles are to be placed at random into the cells of the unit interval which was previously broken into random pieces according to the (Poisson-)Dirichlet partitioning model. Particles being distinguishable, the statistical structure of the problem can be understood within the Maxwell-Boltzmann setup. In this note, we shall address the following sampling problem of a different nature: assume now that indistinguishable particles are to be placed at random within the cells with (Poisson-)Dirichlet distributed sizes. Then the statistical formalism to be used is the one of Bose-Einstein. We show that in the grand canonical ensemble, the Bose sampling procedure from\linebreak (Poisson-)Dirichlet proportions is, to a large extent, amenable to exact analytic calculations. This concerns, for example, the full Bose occupancy distributions, the distribution of the number of distinct occupied fragments, the number of cells with a prescribed amount of particles. Using a grand canonical approach, a phase transition phenomenon is shown to take place provided the disorder parameter of the (Poisson-)Dirichlet partition is large enough; we describe this phase transition in some details.
