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Transactions of the American Mathematical Society

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Markov chains in random environments and random iterated function systems

Author: Örjan Stenflo
Journal: Trans. Amer. Math. Soc. 353 (2001), 3547-3562
MSC (2000): Primary 28A80, 37H99, 60F05, 60J05, 60K37; Secondary 28A78, 60G57, 65C05
Published electronically: April 18, 2001
MathSciNet review: 1837247
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Abstract | References | Similar Articles | Additional Information


We consider random iterated function systems giving rise to Markov chains in random (stationary) environments. Conditions ensuring unique ergodicity and a ``pure type'' characterization of the limiting ``randomly invariant'' probability measure are provided. We also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure.

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  • 1. Barnsley, M. F. and Demko, S. (1985) Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 399, 243-275. MR 87e:58051
  • 2. Billingsley, P. (1968) Convergence of probability measures, Wiley, New York. MR 38:1718
  • 3. Borovkov, A. A. (1998) Ergodicity and stability of stochastic processes, Wiley, Chichester. MR 2000a:60001
  • 4. Borovkov, A. A. and Foss, S. G. (1994) Two ergodicity criteria for stochastically recursive sequences, Acta Appl. Math., 34, 125-134. MR 95g:60083
  • 5. Cogburn, R. (1984) The ergodic theory of Markov chains in random environments, Z. Wahrsch. Verw. Gebiete, 66, 109-128. MR 85k:60100
  • 6. Cogburn, R. (1990) On direct convergence and periodicity for transition probabilities of Markov chains in random environments, Ann. Probab., 18, 642-654. MR 92e:60131
  • 7. Cogburn, R. (1991) On the central limit theorem for Markov chains in random environments Ann. Probab., 19, 587-604. MR 92h:60029
  • 8. Dubins, L. E. and Freedman, D. A. (1966) Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37, 837-848. MR 33:1884
  • 9. Elton, J. H. (1990) A multiplicative ergodic theorem for Lipschitz maps, Stochastic Process. Appl., 34, 39-47. MR 91a:47010
  • 10. Kifer, Y. (1995) Fractals via random iterated function systems and random geometric constructions. Fractal geometry and stochastics (Finsterbergen, 1994), 145-164, Progr. Probab., 37, Birkhäuser, Basel. MR 97j:28013
  • 11. Kifer, Y. (1996) Perron-Frobenius theorem, large deviations, and random perturbations in random environments, Math. Z., 222, 677-698. MR 97f:60131
  • 12. Kifer, Y. (1998) Limit theorems for random transformations and processes in random environments, Trans. Amer. Math. Soc., 350, 1481-1518. MR 98i:60021
  • 13. Letac, G. (1986) A contraction principle for certain Markov chains and its applications, Contemp. Math., 50, 263-273. MR 88a:60121
  • 14. Lu, G. and Mukherjea, A. (1997) Invariant measures and Markov chains with random transition probabilities, Probab. Math. Statist., 17, 115-138. MR 98d:60126
  • 15. Orey, S. (1991) Markov chains with stochastically stationary transition probabilities, Ann. Probab., 19, 907-928. MR 92i:60065
  • 16. Peres, Y., Schlag, W. and Solomyak, B. (2000) Sixty years of Bernoulli convolutions, Fractal geometry and stochastics II (Greifswald, 1998), 39-65, Progr. Probab., 46, Birkhäuser, Basel. CMP 2001:02
  • 17. Propp, J. G. and Wilson, D. B. (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures Algorithms, 9, 223-252. MR 99k:60176
  • 18. Seppäläinen, T. (1994) Large deviations for Markov chains with random transitions, Ann. Probab., 22, 713-748. MR 95j:60044
  • 19. Shiryaev, A. N. (1996) Probability, Second Edition, Springer-Verlag, New York. MR 97c:60003
  • 20. Silvestrov, D. S. and Stenflo, Ö. (1998) Ergodic theorems for iterated function systems controlled by regenerative sequences, J. Theoret. Probab., 11, 589-608. MR 99j:60100
  • 21. Stenflo, Ö. (2001) Ergodic theorems for Markov chains represented by iterated function systems, Bull. Polish Acad. Sci. Math., 49, 27-43.
  • 22. Strichartz, R. S. (1993) Self-similar measures and their Fourier transforms. III., Indiana Univ. Math. J., 42, 367-411. MR 94j:42025
  • 23. Young, L.-S. (1982) Dimension, entropy and Lyapunov exponents. Ergodic Theory Dynam. Systems, 2, 109-124. MR 84k:58087

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Additional Information

Örjan Stenflo
Affiliation: Department of Mathematics, UmeåUniversity, SE-90187 Umeå, Sweden

Keywords: Iterated Function Systems (IFS), Markov chains, pointwise dimension, random environments, exact sampling
Received by editor(s): December 19, 1999
Received by editor(s) in revised form: October 2, 2000
Published electronically: April 18, 2001
Additional Notes: Supported by the The Royal Swedish Academy of Sciences
Article copyright: © Copyright 2001 American Mathematical Society

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