A convenient characterisation of convergent upper transition operators

TL;DR

This paper introduces a new, practical way to characterize the convergence of upper transition operators in imprecise Markov chains, extending the classical ergodicity concept.

Contribution

It provides a general, verifiable condition for convergence based on accessibility and lower reachability, applicable under specific state absorption or finitely generated operators.

Findings

Established a necessary and sufficient condition for convergence.

Connected convergence to accessibility and lower reachability.

Extended the concept of ergodicity to a broader setting.

Abstract

Motivated by its connection to the limit behaviour of imprecise Markov chains, we introduce and study the so-called convergence of upper transition operators: the condition that for any function, the orbit resulting from iterated application of this operator converges. In contrast, the existing notion of `ergodicity' requires convergence of the orbit to a constant. We derive a very general (and practically verifiable) sufficient condition for convergence in terms of accessibility and lower reachability, and prove that this sufficient condition is also necessary whenever (i) all transient states are absorbed or (ii) the upper transition operator is finitely generated.