Electronic Communications in Probability

Some Notes on Topological Recurrence

Niclas Carlsson

Full-text: Open access

Abstract

We review the concept of topological recurrence for weak Feller Markov chains on compact state spaces and explore the implications of this concept for the ergodicity of the processes. We also prove some conditions for existence and uniqueness of invariant measures of certain types. Examples are given from the class of iterated function systems on the real line.

Article information

Source
Electron. Commun. Probab., Volume 10 (2005), paper no. 9, 82-93.

Dates
Accepted: 9 June 2005
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058074

Digital Object Identifier
doi:10.1214/ECP.v10-1137

Mathematical Reviews number (MathSciNet)
MR2150697

Zentralblatt MATH identifier
1108.37008

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Carlsson, Niclas. Some Notes on Topological Recurrence. Electron. Commun. Probab. 10 (2005), paper no. 9, 82--93. doi:10.1214/ECP.v10-1137. https://projecteuclid.org/euclid.ecp/1465058074


Export citation

References

  • M. Barnsley, S. Demko, J. Elton, J. Geronimo. Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. Henri Poincaré, Vol. 24, Iss. 3, pp. 367-394 (1988)
  • P. Billingsley. Convergence of Probability Measures, John Wiley and Sons (1968)
  • L. Breiman. The Strong Law of Large Numbers for a Class of Markov Chains, Annals of Mathematical Statistics, Vol. 31, Iss. 3, pp. 801-803 (1960)
  • L. Breyer, G. Roberts. Catalytic Perfect Simulation, Methodology and Computing in Applied Probability, Vol. 3, Iss. 2, pp. 161-177 (2001)
  • J. Buzzi. Absolutely Continuous S.R.B Measures for Random Lasota-Yorke Maps, Transactions of the American Mathematical Society, Vol. 352, Iss. 7, pp. 3289-3303 (2000)
  • P. Diaconis, D. Freedman. Iterated Random Functions, SIAM Rev., Vol. 41, Iss. 1, pp. 45-76 (1999)
  • L. Dubins, D. Freedman. Invariant Probabilities for Certain Markov Processes, Annals of Mathematical Statistics, Vol. 37, Iss. 4, pp. 837-848 (1966)
  • J. Hobert, C. Robert. A Mixture Representation of $\pi$€ with Applications in Markov Chain Monte Carlo and Perfect Sampling, Annals of Applied Probability, Vol. 14, Iss. 3, pp. 1295-1305 (2004)
  • S. Meyn, R. Tweedie. Markov Chains and Stochastic Stability, Springer Verlag (1993)
  • A. V. Skorokhod. Topologically recurrent Markov chains: Ergodic properties, Theory Probab. Appl, Vol. 31, Iss. 4, pp. 563-571 (1987)
  • H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins, Mathematische Annalen, Vol. 77, pp. 313-352 (1916)
  • K. Yosida. Functional Analysis, Springer Verlag (1968)