Average Behaviour in Discrete-Time Imprecise Markov Chains: A Study of Weak Ergodicity

TL;DR

This paper investigates the weak ergodic behaviour of imprecise Markov chains, establishing conditions for convergence of bounds on expected time averages and demonstrating robustness across different independence assumptions.

Contribution

It introduces a weaker form of ergodicity called weak ergodicity, providing necessary and sufficient conditions for convergence and robustness results independent of independence assumptions.

Findings

Weak ergodicity ensures convergence of bounds regardless of initial state.

Convergence conditions are weaker than traditional ergodicity requirements.

Using actual expected time averages improves approximation over bounds.

Abstract

We study the limit behaviour of upper and lower bounds on expected time averages in imprecise Markov chains; a generalised type of Markov chain where the local dynamics, traditionally characterised by transition probabilities, are now represented by sets of `plausible' transition probabilities. Our first main result is a necessary and sufficient condition under which these upper and lower bounds, called upper and lower expected time averages, will converge as time progresses towards infinity to limit values that do not depend on the process' initial state. Our condition is considerably weaker than that needed for ergodic behaviour; a similar notion which demands that marginal upper and lower expectations of functions at a single time instant converge to so-called limit-or steady state-upper and lower expectations. For this reason, we refer to our notion as `weak ergodicity'. Our second main result shows that, as far as this weakly ergodic behaviour is concerned, one should not worry about which type of independence assumption to adopt-epistemic irrelevance, complete independence or repetition independence. The characterisation of weak ergodicity as well as the limit values of upper and lower expected time averages do not depend on such a choice. Notably, this type of robustness is not exhibited by the notion of ergodicity and the related inferences of limit upper and lower expectations. Finally, though limit upper and lower expectations are often used to provide approximate information about the limit behaviour of time averages, we show that such an approximation is sub-optimal and that it can be significantly improved by directly using upper and lower expected time averages.