Electronic Communications in Probability

On the Zero-One Law and the Law of Large Numbers for Random Walk in Mixing Random Environment

Firas Rassoul-Agha

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Abstract

We prove a weak version of the law of large numbers for multi-dimensional finite range random walks in certain mixing elliptic random environments. This already improves previously existing results, where a law of large numbers was known only under strong enough transience. We also prove that for such walks the zero-one law implies a law of large numbers.

Article information

Source
Electron. Commun. Probab., Volume 10 (2005), paper no. 5, 36-44.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058070

Digital Object Identifier
doi:10.1214/ECP.v10-1130

Mathematical Reviews number (MathSciNet)
MR2119152

Zentralblatt MATH identifier
1060.60101

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

Citation

Rassoul-Agha, Firas. On the Zero-One Law and the Law of Large Numbers for Random Walk in Mixing Random Environment. Electron. Commun. Probab. 10 (2005), paper no. 5, 36--44. doi:10.1214/ECP.v10-1130. https://projecteuclid.org/euclid.ecp/1465058070


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References

  • Alili, S. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 (1999), no. 2, 334–349.
  • Comets, F. and Zeitouni, O. A law of large numbers for random walks in random mixing environments. Ann. Probab. 32 (2004), no. 1B, 880–914.
  • Dobrushin, R. L. and Shlosman, S. B. Completely analytical Gibbs fields. Statistical physics and dynamical systems (Köszeg, 1984), 371–403, Progr. Phys., 10, Birkhäuser Boston, Boston, MA, 1985.
  • Georgii, H.-O. Gibbs measures and phase transitions. de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 1988. xiv+525 pp. ISBN: 0-89925-462-4
  • Kalikow, S. A. Generalized random walk in a random environment. Ann. Probab. 9 (1981), no. 5, 753–768.
  • Rassoul-Agha, F. The point of view of the particle on the law of large numbers for random walks in a mixing random environment. Ann. Probab. 31 (2003), no. 3, 1441–1463.
  • Rassoul-Agha, F. Large deviations for random walks in a mixing random environment and other (non-Markov) random walks. Comm. Pure Appl. Math. 57 (2004), no. 9, 1178–1196.
  • Solomon, F. Random walks in a random environment. Ann. Probability 3 (1975), 1–31.
  • Sznitman, A.-S. An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 (2002), no. 4, 509–544.
  • Sznitman, A.-S. and Zerner, M. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999), no. 4, 1851–1869.
  • Varadhan, S. R. S. Large deviations for random walks in a random environment. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56 (2003), no. 8, 1222–1245.
  • Zerner, M. P. W. A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Comm. Probab. 7 (2002), 191–197 (electronic).
  • Zerner, M. P. W. and Merkl, F. A zero-one law for planar random walks in random environment. Ann. Probab. 29 (2001), no. 4, 1716–1732.