ASYMPTOTIC PROPERTIES WITH PROBABILITY 1 FOR ONE-DIMENSIONAL RANDOM WALKS IN A RANDOM ENVIRONMENT

Abstract

Random walks in a random environment are considered on the set of integers when the moving particle can go at most steps to the right and at most steps to the left in a unit of time. The transition probabilities for such a random walk from a point are determined by the vector . It is assumed that the sequence is a sequence of independent identically distributed random vectors. Asymptotic properties with probability 1 are investigated for such a random process. An invariance principle and the law of the iterated logarithm for a product of independent random matrices are proved as auxiliary results.