Abstract
We study ergodicity for upper transition operators: bounded, sub-additive and non-negatively homogeneous transformations of finite-dimensional linear spaces. Ergodicity provides a necessary and sufficient condition for Perron–Frobenius-like convergence behaviour for upper transition operators. It can also be characterised alternatively using accessibility relations: ergodicity is equivalent with there being a single maximal communication (or top) class that is moreover regular and absorbing. We present efficient algorithms for checking these conditions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
De Cooman, G., Hermans, F., Quaeghebeur, E.: Imprecise Markov chains and their limit behavior. Probability in the Engineering and Informational Sciences 23(4), 597–635 (2009)
Hartfiel, D.J.: Markov Set-Chains. Lecture Notes in Mathematics, vol. 1695. Springer, Berlin (1998)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Undergraduate Text in Mathematics. Springer, New York (1976); reprint of the 1960 Edition
Lemmens, B., Scheutzow, M.: On the dynamics of sup-norm non-expansive maps. Ergod. Theor. Dyn. Syst. 25, 861–871 (2005)
Seneta, E.: Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, Heidelberg (2006)
Sine, R.: A nonlinear Perron–Frobenius theorem. Proceedings of the American Mathematical Society 109(2), 331–336 (1990)
Škulj, D., Hable, R.: Coefficients of ergodicity for imprecise markov chains. In: Augustin, T., Coolen, F.P., Serafin, M., Troffaes, M.C. (eds.) ISIPTA 2009 – Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications, pp. 377–386. SIPTA (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hermans, F., de Cooman, G. (2010). Ergodicity Conditions for Upper Transition Operators. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods. IPMU 2010. Communications in Computer and Information Science, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14055-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-14055-6_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14054-9
Online ISBN: 978-3-642-14055-6
eBook Packages: Computer ScienceComputer Science (R0)