Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators

Małgorzata Pułka

Discussiones Mathematicae Probability and Statistics (2012)

  • Volume: 32, Issue: 1-2, page 17-33
  • ISSN: 1509-9423

Abstract

top
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.

How to cite

top

Małgorzata Pułka. "Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators." Discussiones Mathematicae Probability and Statistics 32.1-2 (2012): 17-33. <http://eudml.org/doc/270869>.

@article{MałgorzataPułka2012,
abstract = {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.},
author = {Małgorzata Pułka},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {Markov operator; asymptotic stability; residuality; dense $G_\{δ\}$; denseness},
language = {eng},
number = {1-2},
pages = {17-33},
title = {Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators},
url = {http://eudml.org/doc/270869},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Małgorzata Pułka
TI - Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators
JO - Discussiones Mathematicae Probability and Statistics
PY - 2012
VL - 32
IS - 1-2
SP - 17
EP - 33
AB - 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.
LA - eng
KW - Markov operator; asymptotic stability; residuality; dense $G_{δ}$; denseness
UR - http://eudml.org/doc/270869
ER -

References

top
  1. [1] Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices, Ann. Polon. Math. 52 (1990) 165-173 Zbl0719.47022
  2. [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 Zbl1093.47009
  3. [3] W. Bartoszek and M. Pułka, On mixing in the class of quadratic stochastic operators, submitted to Nonlinear Anal. Theory Methods Appl. Zbl1298.47069
  4. [4] More on the 'zero-two' law, Proc. Amer. Math. Soc 61 (1976) 262-264 
  5. [5] On ergodic properties of inhomogeneous Markov processes, Rev. Roumaine Math. Pures Appl. 43 (1998) 375-392 Zbl0998.60067
  6. [6] On two recent papers on ergodicity in nonhomogeneous Markov chains, Annals Math. Stat. 43 (1972) 1732-1736. doi: 10.1214/aoms/1177692411 Zbl0249.60031
  7. [7] Finite Markov Processes and Their Applications (John Wiley and Sons, 1980). 
  8. [8] Markov Chains: Theory and Applications (Wiley, New York, 1976). 
  9. [9] Structure of mixing and category of complete mixing for stochastic operators, Ann. Polon. Math. 56 (1992) 233-242 Zbl0786.47004
  10. [10] Markov Chains and Stochastic Stability (Springer, London, 1993). doi: 10.1007/978-1-4471-3267-7 
  11. [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 Zbl1334.60145
  12. [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 Zbl1213.60120
  13. [13] Most Markov operators on C(X) are quasi-compact and uniquely ergodic, Colloq. Math. 52 (1987) 277-280 Zbl0624.47021

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.