Abstract
We study the behavior of persistent random walks (RW) on the integers in a random environment. A complete characterization of the almost sure limit behavior of these processes, including the law of large numbers, is obtained. This is done in a general situation where the environmental sequence of random variables is stationary and ergodic. Szász and Tóth obtained a central limit theorem when the ratio μ/λ, of right- and left-transpassing probabilities satisfies μ/λ≤a<1 a.s. (for a given constant a). We consider the case where μ/λ has wider fluctuations; we shall observe that an unusual situation arises: the RW may converge a.s. to infinity even with zero drift. Then, we obtain nonclassical limiting distributions for the RW. Proofs are based on the introduction of suitable branching processes in order to count the steps performed by the RW.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
K. B. Athreya and S. Karlin, On Branching Processes with random environments I, Annals of Mathematical Statistics 42(5):1499–1520 (1971).
K. B. Athreya and S. Karlin, On Branching Processes with random environments II, Annals of Mathematical Statistics 42(6):1843–1858 (1971).
K. B. Athreya and P. E. Ney, Branching Processes (Springer-Verlag, Berlin/Heidelberg/New York, 1972).
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 3rd ed., John Wiley & Sons, New York.
B. V. Gnedenko and P. E. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison Wesley Pub. Co., 1954).
Y. Guivarc'h et A. Raugi, Frontière de Furstunberg, propriétés de contraction et théorèmes de convergence, Z. Wahrs. Geb. 69:187–242 (1987).
T. Harris, First passage and recurrence distributions, Trans. Amer. Math. Soc. 73:471–486 (1952).
H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Mathematica 131:208–248 (1973).
H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in random environment, Compositio Mathematica 30(2):145–168 (1975).
E. Key, Limiting Distributions and Regeneration Times for Multitype Branching Processes with Immigration in a Random Environment, Annals of Probability 15(1):344–353 (1987).
S. M. Kozlov, The method of averaging and walks in inhomogenious environments, Russian Math. Surveys 40(2):73–145 (1985).
D. Szász and B. Tóth, Persistent random walks in one-dimensional random environment, Journal of Statistical Physics 37(1/2):27–38 (1984).
Ya. G. Sinai, Limit behaviour of one dimensional random walk in a random environment, Teoriya Veroyat. i Ee Primeneniya 27–2:247–258 (1982).
F. Solomon, Random walks in a random environment, Annals of Probability 8(1):1–31 (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alili, S. Persistent Random Walks in Stationary Environment. Journal of Statistical Physics 94, 469–494 (1999). https://doi.org/10.1023/A:1004596204224
Issue Date:
DOI: https://doi.org/10.1023/A:1004596204224