Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 2, 162--189

Statistics of Bose Samples from Dirichlet Proportions

Thierry Huillet, Universit\'{e} de Cergy-Pontoise, France

SUMMARY. 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.

AMS (2000) subject classification. Primary 60G57; secondary 60K99, 62E15, 62E17.

Key words and phrases. Random discrete distribution, Dirichlet, sampling, Ewens, urns, Maxwell-Boltzmann and Bose-Einstein statistics, disordered systems, phase transition.

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