Characterising the Convergence of Imprecise Markov Chains

Jasper De Bock, Alexander Erreygers, Floris Persiau
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:620-639, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-de-bock26a, title = {Characterising the Convergence of Imprecise {Markov} Chains}, author = {De Bock, Jasper and Erreygers, Alexander and Persiau, Floris}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {620--639}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/de-bock26a/de-bock26a.pdf}, url = {https://proceedings.mlr.press/v337/de-bock26a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Characterising the Convergence of Imprecise Markov Chains %A Jasper De Bock %A Alexander Erreygers %A Floris Persiau %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-de-bock26a %I PMLR %P 620--639 %U https://proceedings.mlr.press/v337/de-bock26a.html %V 337 %X 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.
APA
De Bock, J., Erreygers, A. & Persiau, F.. (2026). Characterising the Convergence of Imprecise Markov Chains. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:620-639 Available from https://proceedings.mlr.press/v337/de-bock26a.html.

Related Material