Limit theorems for one-dimensional transient random walks in Markov environments. (English) Zbl 1070.60024

Mayer-Wolf, Eddy; Roitershtein, Alexander; Zeitouni, Ofer

Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 5, 635-659 (2004).

The authors consider a one-dimensional random walk in random environment, which is given as a function of a stationary Markov chain. They classify all the possible limits in distribution (averaged both over the process and the environment). This extends results of H. Kesten, M. V. Kozlov and F. Spitzer [Compos. Math. 30, 145–168 (1975; Zbl 0388.60069)] for the case of an i.i.d. environment. As in the above-mentioned work, there are different stable limit laws as well as a regime where the classical central limit theorem holds. The Markov chain defining the environment is assumed to be irreducible, with transition kernel dominated above and below by some probability measure. The main idea of the proof is, similar to the case of an i.i.d. environment, to analyse the hitting times and to use regeneration techniques.

Reviewer: Nina Gantert (Münster)

MSC:

60F05	Central limit and other weak theorems
60K37	Processes in random environments
60J05	Discrete-time Markov processes on general state spaces
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

RWRE; stable laws; branching; stochastic difference equations