Finite sample properties of the mean occupancy counts and probabilities

Geoffrey Decrouez, Michael Grabchak, Quentin Paris

Published 2016-01-25Version 1

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 $\mathbb E M_{n,r}$. In particular, both upper and lower bounds are given in terms of the right tail $\nu$ of the counting measure 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 confidence regions. At the end of the paper, we discuss an extension of the occupancy problem to arbitrary distributions in a metric space.