Open Access
August 2018 Finite sample properties of the mean occupancy counts and probabilities
Geoffrey Decrouez, Michael Grabchak, Quentin Paris
Bernoulli 24(3): 1910-1941 (August 2018). DOI: 10.3150/16-BEJ915


For a probability distribution $P$ on an at most countable alphabet $\mathcal{A}$, this article gives finite sample bounds for the expected occupancy counts $\mathbb{E}K_{n,r}$ and probabilities $\mathbb{E}M_{n,r}$. Both upper and lower bounds are given in terms of the counting function $\nu$ of $P$. Special attention is given to the case where $\nu$ is bounded by a regularly varying function. In this case, it is shown that our general results lead to an optimal-rate control of the expected occupancy counts and probabilities with explicit constants. Our results are also put in perspective with Turing’s formula and recent concentration bounds to deduce bounds in probability. At the end of the paper, we discuss an extension of the occupancy problem to arbitrary distributions in a metric space.


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Geoffrey Decrouez. Michael Grabchak. Quentin Paris. "Finite sample properties of the mean occupancy counts and probabilities." Bernoulli 24 (3) 1910 - 1941, August 2018.


Received: 1 February 2016; Revised: 1 October 2016; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839255
MathSciNet: MR3757518
Digital Object Identifier: 10.3150/16-BEJ915

Keywords: counting measure , finite sample bounds , occupancy problem , regular variation , Turing’s formula , urn scheme

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
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