Characterising the Convergence of Imprecise Markov Chains

Motivated by their connection to the limit behaviour of imprecise {Markov} chains, we study the asymptotic behaviour of upper transition operators. Our focus is on their convergence: the condition that for every real function, the sequence generated by repeated application of the operator admits a well-defined limit. This notion is strictly weaker than classical ergodicity, which enforces convergence to a constant, and therefore requires a different analytic treatment. We develop a full characterisation of convergence in terms of graph-theoretic relations induced by the operator: accessibility and lower reachability. The resulting criterion is both necessary and sufficient, and applies to arbitrary upper transition operators without structural restrictions, thus strengthening earlier work that provided only sufficient conditions for the unrestricted case.