Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip

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).