Abstract
In Refs. 7, 8, 12–14, applications of Birkhoff's coefficient of ergodicity to the study of products of finite and infinite matrices are given. In this paper we intend to show that using the ideas and results of Birkhoff and Baueret al. one can establish such results in the general setting of positive operators on a space of measures and in particular, in the study of nonhomogeneous Markov Chains in general state spaces. In this setup the proofs are very simple. Since the high level of generality in Ref. 1 hinders an easy understanding of the special case of interest to probabilists, we present a self-contained treatment. Even though our aim is methodological, our theorems are new, even in the case of infinite matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauer, F. L., Deutsch, E. and Stoer, J. (1969). Abschätzungen für die Eigenwerte positiver linearer Operatoren,Linear Algebra and its Applications 2, 275–301.
Birkhoff, G. (1967). Lattice theory.Amer. Math. Soc. Coll. Publcns., XXV.
Bushell, P. J. (1973). Hilbert's Metric and Positive Contraction Mapping in a Banach Space,Arch. Rational Mech. Anal. 52, 330–338.
Dobrushin, R. L. (1956). Central limit theorem for nonstationary Markov chains. 1, 2.,Theory Prob. Appl. 1, 65–80, 329–384.
Fleischer, I., and Joffe, A. (1977). Perturbation des produits infinis et applications,Journal d'Analyse Math. 31, 69–75.
Fleischer, I., and Joffe, A. (1993). Behavior of infinite products with applications to non-homogeneous Markov Chains,Contemporary Mathematics: Doeblin and Modern Probability. Am. Math. Soc. 149, 179–187.
Gibert, S., and Mukherjea, A. (1981). Products of Infinite-Dimensional Non-Negative Matrices; Extensions of Hajnal's Results,Math Z. 196, 485–490.
Hajnal, J. (1976). On products of non-negative matrices,Math. Proc. Camb. Philos. Soc. 79, 521–530.
Krasnosel'skij, M. A., Lifshits, Je. A. and Sobolev, A. V. (1989).Positive Linear Systems—The Method of Positive Operators, Heldermann Verlag, Berlin.
Ostrowski, A. M. (1964).Positive matrices and Functional analysis. Recent Advances in Matrix Theory, University of Wisconsin Press, Madison.
Schaefer, H. H. (1971).Topological Vector Spaces, Springer Verlag, Second ed.
Seneta, E. (1979). Coefficients of Ergodicity: Structure and Applications,Adv. Appl. Prob. 11, 576–590.
Seneta, E. (1981).Non-negative Matrices and Markov Chains, Springer Verlag, Second. ed.
Seneta, E. (1993). Ergodicity for Products of Infinite stochastic Matrices, Manuscript Feb. 10.
Author information
Authors and Affiliations
Additional information
Support from the NSERC grant of Professor Rosenberg is gratefully acknowledged.
Research supported by an NSERC grant.
Rights and permissions
About this article
Cite this article
Fleischer, I., Joffe, A. Ratio ergodicity for non-homogeneous Markov Chains in general state spaces. J Theor Probab 8, 31–37 (1995). https://doi.org/10.1007/BF02213452
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02213452