## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 1910-1941.

### Finite sample properties of the mean occupancy counts and probabilities

Geoffrey Decrouez, Michael Grabchak, and Quentin Paris

#### 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 3 (2018), 1910-1941.

**Dates**

Received: February 2016

Revised: October 2016

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540463

**Digital Object Identifier**

doi:10.3150/16-BEJ915

**Mathematical Reviews number (MathSciNet)**

MR3757518

**Zentralblatt MATH identifier**

06839255

**Keywords**

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

#### Citation

Decrouez, Geoffrey; Grabchak, Michael; Paris, Quentin. Finite sample properties of the mean occupancy counts and probabilities. Bernoulli 24 (2018), no. 3, 1910--1941. doi:10.3150/16-BEJ915. https://projecteuclid.org/euclid.bj/1517540463