Occupancy distributions in Markov chains via Doeblin's ergodicity coefficient

TL;DR

This paper introduces a novel method using Doeblin's ergodicity coefficient to approximate the occupancy distribution in finite Markov chains, especially useful when exact calculations are infeasible.

Contribution

The paper develops a new approximation technique leveraging Doeblin's coefficient, enabling efficient analysis of occupancy distributions in non-stationary Markov chains.

Findings

Provides a practical approximation method for occupancy distributions

Applicable to non-stationary and complex Markov chains

Useful for pattern detection in Markovian and non-Markovian sequences

Abstract

We apply Doeblin's ergodicity coefficient as a computational tool to approximate the occupancy distribution of a set of states in a homogeneous but possibly non-stationary finite Markov chain. Our approximation is based on new properties satisfied by this coefficient, which allow us to approximate a chain of duration n by independent and short-lived realizations of an auxiliary homogeneous Markov chain of duration of order ln(n). Our approximation may be particularly useful when exact calculations via first-step methods or transfer matrices are impractical, and asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques.