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2012 | 32 | 1-2 | 17-33
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Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators

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EN
Abstrakty
EN
We study different types of asymptotic behaviour in the set of (infinite dimensional) nonhomogeneous chains of stochastic operators acting on L¹(μ) spaces. In order to examine its structure we consider different norm and strong operator topologies. To describe the nature of the set of nonhomogeneous chains of Markov operators with a particular limit behaviour we use the category theorem of Baire. We show that the geometric structure of the set of those stochastic operators which have asymptotically stationary density differs depending on the considered topologies.
Twórcy
  • Department of Mathematics, Gdańsk University of Technology, ul. Gabriela Narutowicza 11/12, 80-233, Gdańsk, Poland
Bibliografia
  • [1] Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices, Ann. Polon. Math. 52 (1990) 165-173
  • [2] On residualities in the set of Markov operators on 𝓒₁, Proc. Amer. Math. Soc. 133 (2005) 2119-2129. doi: 10.1090/S0002-9939-05-07776-2
  • [3] W. Bartoszek and M. Pułka, On mixing in the class of quadratic stochastic operators, submitted to Nonlinear Anal. Theory Methods Appl.
  • [4] More on the 'zero-two' law, Proc. Amer. Math. Soc 61 (1976) 262-264
  • [5] On ergodic properties of inhomogeneous Markov processes, Rev. Roumaine Math. Pures Appl. 43 (1998) 375-392
  • [6] On two recent papers on ergodicity in nonhomogeneous Markov chains, Annals Math. Stat. 43 (1972) 1732-1736. doi: 10.1214/aoms/1177692411
  • [7] Finite Markov Processes and Their Applications (John Wiley and Sons, 1980).
  • [8] Markov Chains: Theory and Applications (Wiley, New York, 1976).
  • [9] Structure of mixing and category of complete mixing for stochastic operators, Ann. Polon. Math. 56 (1992) 233-242
  • [10] Markov Chains and Stochastic Stability (Springer, London, 1993). doi: 10.1007/978-1-4471-3267-7
  • [11] F. Mukhamedov, On L₁-weak ergodicity of nonhomogeneous discrete Markov processes and its applications, Rev. Mat. Complut., in press. doi: 10.1007/s13163-012-0096-9
  • [12] On the mixing property and the ergodic principle for nonhomogeneous Markov chains, Linear Alg. Appl. 434 (2011) 1475-1488. doi: 10.1016/j.laa.2010.11.021
  • [13] Most Markov operators on C(X) are quasi-compact and uniquely ergodic, Colloq. Math. 52 (1987) 277-280
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1141
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