Finite sample properties of the mean occupancy counts and probabilities

TL;DR

This paper derives finite sample bounds for expected occupancy counts and probabilities for countable alphabets, with special focus on regularly varying distributions, providing optimal-rate controls and connections to concentration bounds.

Contribution

It introduces new finite sample bounds for occupancy metrics, especially for distributions with regularly varying counting functions, enhancing understanding of their probabilistic behavior.

Findings

Derived finite sample bounds for expected occupancy counts and probabilities.

Established optimal-rate control results for regularly varying distributions.

Connected occupancy bounds with concentration inequalities and extended to metric spaces.

Abstract

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.