Some Notes on Topological Recurrence

Niclas Carlsson (Abo Akademi University, Finland)

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.

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Pages: 82-93

Publication Date: June 9, 2005

DOI: 10.1214/ECP.v10-1137

References

  1. 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) Math. Review 89k:60088
  2. P. Billingsley. Convergence of Probability Measures, John Wiley and Sons (1968) Math. Review 38 #1718
  3. 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) Math. Review 22 #8560
  4. L. Breyer, G. Roberts. Catalytic Perfect Simulation, Methodology and Computing in Applied Probability, Vol. 3, Iss. 2, pp. 161-177 (2001) Math. Review 2002i:60103
  5. 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) Math. Review 2001a:37035
  6. P. Diaconis, D. Freedman. Iterated Random Functions, SIAM Rev., Vol. 41, Iss. 1, pp. 45-76 (1999) Math. Review 2000c:60102
  7. L. Dubins, D. Freedman. Invariant Probabilities for Certain Markov Processes, Annals of Mathematical Statistics, Vol. 37, Iss. 4, pp. 837-848 (1966) Math. Review 33 #1884
  8. J. Hobert, C. Robert. A Mixture Representation of π with Applications in Markov Chain Monte Carlo and Perfect Sampling, Annals of Applied Probability, Vol. 14, Iss. 3, pp. 1295-1305 (2004) Math. Review 2005d:60106
  9. S. Meyn, R. Tweedie. Markov Chains and Stochastic Stability, Springer Verlag (1993) Math. Review 95j:60103
  10. A. V. Skorokhod. Topologically recurrent Markov chains: Ergodic properties, Theory Probab. Appl, Vol. 31, Iss. 4, pp. 563-571 (1987) Math. Review 88i:60110
  11. H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins, Mathematische Annalen, Vol. 77, pp. 313-352 (1916) Math. Review number not available.
  12. K. Yosida. Functional Analysis, Springer Verlag (1968) Math. Review 39 #741
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