Abstract
We extend an inequality (which involves the Dobrushin coefficient of ergodicity; see Cohen et al.(4)) to any linear bounded operator with domain and codomain L 1-spaces. We use the extended Dobrushin coefficient of ergodicity, that appears in the inequality, in order to obtain sufficient conditions for the uniform asymptotic stability of a positive contraction of an L 1-space. We conclude the paper by studying a class of strongly asymptotically stable positive contractions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Akcoglu, M., and Sucheston, L. (1972). On operator convergence in Hilbert space and in Lebesgue space. Period. Math. Hungar. 2, 235–244.
Aliprantis, C. D., and Burkinshaw, O. (1985). Positive Operators, Academic Press, New York/London.
Bartoszek, W. (1990). Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices. Annal. Polon. Math. 52, 165–173.
Cohen, J. E., Iwasa, Y., Rautu, G., Ruskai, M. B., Seneta, E., and Zbaganu, G. (1993). Relative entropy under mappings by stochastic matrices. Linear Algebra Appl. 179, 211–235.
Derriennic, Y. (1976). Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré, Probab. Statist. 12B, 111–129.
Dobrushin, R. L. (1956). Central limit theorem for nonstationary Markov chains. I and II. Th. Probab. Appl. 1, 65–80; 329–383.
Dunford, N. and Schwartz, J. T. (1958). Linear Operators, Part I: General Theory, Interscience, New York.
Halmos, P. R. (1974). Measure Theory, Springer, New York.
Komorowski, T., and Tyrcha, J. (1989). Asymptotic properties of some Markov operators. Bull. Polish Acad. Sci. Math. 37, 220–228.
Krengel, U. (1985). Ergodic Theorems, Walter de Gruyter, Berlin/New York.
Krengel, U., and Sucheston, L. (1969). On mixing in infinite measure spaces. Z. Wahrsch. verw. Geb. 13, 150–164.
Lin, M. (1971). Mixing for Markov operators. Z. Wahrsch. Verw. Geb. 19, 231–242.
Neveu, J. (1965). Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, London, Amsterdam.
Ornstein, D., and Sucheston, L. (1970). An operator theorem on L 1 convergence to zero with applications to Markov kernels. Ann. Math. Statist. 41, 1631–1639.
Räbiger, F. (1993). Stability and ergodicity of dominated semigroups. I. The uniform case. Math. Z. 214, 43–54.
Revuz, D. (1975). Markov Chains, North-Holland, Amsterdam/Oxford.
Schaefer, H. H. (1974). Banach Lattices and Positive Operators, Springer, New York/Heidelberg/Berlin.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zaharopol, R., Zbaganu, G. Dobrushin Coefficients of Ergodicity and Asymptotically Stable L 1-Contractions. Journal of Theoretical Probability 12, 885–902 (1999). https://doi.org/10.1023/A:1021684818286
Issue Date:
DOI: https://doi.org/10.1023/A:1021684818286