The Annals of Applied Probability

Bins and balls; Large deviations of the empirical occupancy process

Stéphane Boucheron, Fabrice Gamboa, and Christian Léonard

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Abstract

In the random allocation model, balls are sequentially inserted at random into $n$ exchangeable bins. The occupancy score of a bin denotes the number of balls inserted in this bin. The (random) distribution of occupancy scores defines the object of this paper: the empirical occupancy measure which is a probability measure over the integers. This measure-valued random variable packages many useful statistics. This paper characterizes the large deviations of the flow of empirical occupancy measures when $n$ goes to infinity while the number of inserted balls remains proportional to $n$. The main result is a Sanov-like theorem for the empirical occupancy measure when the set of probability measures over the integers is endowed with metrics that are slightly stronger than the total variation distance. Thanks to a coupling argument, this result applies to the degree distribution of sparse random graphs.

Article information

Source
Ann. Appl. Probab., Volume 12, Number 2 (2002), 607-636.

Dates
First available in Project Euclid: 17 July 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1026915618

Digital Object Identifier
doi:10.1214/aoap/1026915618

Mathematical Reviews number (MathSciNet)
MR1910642

Zentralblatt MATH identifier
1013.60017

Subjects
Primary: 60F10: Large deviations 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 05C40: Connectivity
Secondary: 05C80: Random graphs [See also 60B20] 05A99: None of the above, but in this section

Keywords
Random graphs large deviations occupancy Orlicz spaces Sanov theorem Poisson approximation

Citation

Boucheron, Stéphane; Gamboa, Fabrice; Léonard, Christian. Bins and balls; Large deviations of the empirical occupancy process. Ann. Appl. Probab. 12 (2002), no. 2, 607--636. doi:10.1214/aoap/1026915618. https://projecteuclid.org/euclid.aoap/1026915618


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  • AND ECOLE POLy TECHNIQUE, CMAP 91128 PALAISEAU FRANCE E-MAIL: christian.leonard@u-paris10.fr