A convenient characterisation of convergent upper transition operators

Jasper De Bock, Alexander Erreygers, Floris Persiau
Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 290:115-125, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v290-de-bock25a, title = {A convenient characterisation of convergent upper transition operators}, author = {De~Bock, Jasper and Erreygers, Alexander and Persiau, Floris}, booktitle = {Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications}, pages = {115--125}, year = {2025}, editor = {Destercke, Sébastien and Erreygers, Alexander and Nendel, Max and Riedel, Frank and Troffaes, Matthias C. M.}, volume = {290}, series = {Proceedings of Machine Learning Research}, month = {15--18 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v290/main/assets/de-bock25a/de-bock25a.pdf}, url = {https://proceedings.mlr.press/v290/de-bock25a.html}, 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.} }
Endnote
%0 Conference Paper %T A convenient characterisation of convergent upper transition operators %A Jasper De Bock %A Alexander Erreygers %A Floris Persiau %B Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications %C Proceedings of Machine Learning Research %D 2025 %E Sébastien Destercke %E Alexander Erreygers %E Max Nendel %E Frank Riedel %E Matthias C. M. Troffaes %F pmlr-v290-de-bock25a %I PMLR %P 115--125 %U https://proceedings.mlr.press/v290/de-bock25a.html %V 290 %X 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.
APA
De Bock, J., Erreygers, A. & Persiau, F.. (2025). A convenient characterisation of convergent upper transition operators. Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications, in Proceedings of Machine Learning Research 290:115-125 Available from https://proceedings.mlr.press/v290/de-bock25a.html.

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