Abstract
The main goal of this work is to study the asymptotic behaviour of hitting times of a random walk (RW) in a quenched random environment (RE) on a strip. We introduce enlarged random environments in which the traditional hitting time can be presented as a sum of independent random variables whose distribution functions form a stationary random sequence. This allows us to obtain conditions (stated in terms of properties of random environments) for a linear growth of hitting times of relevant random walks. In some important cases (e.g. independent random environments) these conditions are also necessary for this type of behaviour. We also prove the quenched Central Limit Theorem (CLT) for hitting times in the general ergodic setting. A particular feature of these (ballistic) laws in random environment is that, whenever they hold under standard normalization, the convergence is a convergence with a speed. The latter is due to certain properties of moments of hitting times which are also studied in this paper. The asymptotic properties of the position of the walk are stated but are not proved in this work since this has been done in Goldhseid (Probab. Theory Relat. Fields 139(1):41–64, 2007).
References
Alili S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36: 334–349
Atkinson F.V. (1964). Discrete and Continuous Boundary Problems. Academic, New York
Bolthausen E. and Goldsheid I. (2000). Recurrence and transience of random walks in random environments on a strip. Commun. Math. Phys. 214: 429–447
Bolthausen E. and Sznitman A. (2002). Ten lectures on Random Media, DMV-Lectures, vol. 32. Birkhäuser, Basel
Brémont J. (2002). On some random walks on \({\mathbb{Z}}\) in random medium Ann. Probab. 30: 1266–1312
Brémont J. (2004). Random walks on \({\mathbb{Z}}\) in random medium and Lyapunov spectrum Ann. Inst. H. Poincare Prob/Stat 40: 309–336
Derriennic Y. (1999). Sur la récurrence des marches aléatoires unidimensionnelles en environement aléatoire. C. R. Acad. Sci. Paris Sér. I Math. 329(1): 65–70
Furstenberg H. and Kesten H. (1960). Products of random matrices. Ann. Math. Stat. 31: 457–469
Goldhseid I. (2007). Simple transient random walks in one-dimensional random environment. Probab. Theory Relat. Fields 139(1): 41–64
Guivarc’h, Y., Le Page, E.: Simplicité de spectres de Lyapunov et propriété d’isolation spectrale pour une famille d’opérateurs sur l’espace projectif. In: Kaimanovitch, V. (ed.) Random walks and Geometry, pp. 181–259, De Gruyter (2004)
Kesten H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131: 207–248
Kesten H., Kozlov M.V. and Spitzer F. (1975). Limit law for random walk in a random environment. Compos. Math. 30: 145–168
Key E. (1984). Recurrence and transience criteria for a random walk in random environment. Ann. Probab. 12: 529–560
Kifer Y. (1996). Perron-Frobenius theorem, large deviations and random pertubations in random environments. Math. Zeitschrift 222: 677–698
Kifer Y. (1998). Limit theorems for random transformations and processes in random environments. Trans. Am. Math. Soc. 348: 2003–2038
Kozlov M.V. (1973). A random walk on a line with stochastic structure (in Russian). Probab. Theory Appl. 18: 406–408
Ledrappier, F.: Quelques propriétés des exposants caractéristiques, Ecole d‘Eté de Saint-Flour 1982, Lecture Notes in Mathematics, vol. 1097, pp. 305–396. Springer, Berlin (1984)
Letchikov A.V. (1993). A criterion for linear drift and the central limit theorem for one-dimensional random walks in a random environment. Russ. Acad. Sci. Sb. Math. 79: 73–92
Mayer-Wolf E., Roitershtein A. and Zeitouni O. (2004). Limit theorems for one-dimensional random walks in Markov random environments. Ann. Inst. H. Poincare Prob/Stat 40: 635–659
Molchanov, S.A.: Lectures on random media. Ecole d’Eté de Prbabilités de St. Flour XXII-1992. Lecture Notes in Mathematics, vol. 1581. Springer, Berlin (1994)
Roitershtein, A.: Tranzient random walks on a strip in a random environment. Preprint (2006)
Sinai Ya.G. (1982). The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27: 256–268
Sinai Ya.G. (1999). Simple random walk on tori. J. Stat. Phys. 94(3–4): 695–708
Solomon F. (1975). Random walks in a random environment. Ann. Probab. 3: 1–31
Sznitman A.-S. and Zerner M.P.W. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27(4): 1851–1869
Sznitman A.-S. (2000). Slowdown and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2: 93–143
Sznitman A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Relat. Fields 122(4): 509–544
Sznitman, A.-S.: Topics in random walks in random environment. In: School and Conference on Probability Theory, ICTP Lecture Notes Series, Trieste, vol. 17, pp. 203–266 (2004)
Zeitouni, O.: Random walks in random environment, XXXI Summer school in Probability, St. Flour (2001). Lecture notes in Mathematics, vol. 1837, pp. 193–312. Springer, Berlin (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goldsheid, I.Y. Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip. Probab. Theory Relat. Fields 141, 471–511 (2008). https://doi.org/10.1007/s00440-007-0091-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0091-0
Keywords
- RWRE
- Random walks on a strip
- Quenched random environments
- Central limit theorem
Mathematics Subject Classification (2000)
- Primary: 60K37
- 60F05
- Secondary: 60J05
- 82C44