# On the occupancy problem for a regime switching model

## TL;DR

This paper investigates occupancy probabilities in a regime switching Markov chain model, providing finite sample bounds and asymptotic analysis, especially for regularly varying distributions, revealing rate optimal bounds.

## Contribution

It introduces finite sample bounds and asymptotic results for occupancy probabilities in a regime switching Markov chain, extending beyond iid assumptions.

## Key findings

- Finite sample bounds are rate optimal.
- Asymptotic behavior characterized for regularly varying distributions.
- Bounds decay at the same rate as asymptotics.

## Abstract

This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed (iid), we assume that they follow a regime switching Markov chain. For this model, we 1) give finite sample bounds on the occupancy probabilities, and 2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that, in the regularly varying case, the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.