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

Random walks in a random environment are considered on the set Z of integers when the moving particle can go at most R steps to the right and at most L steps to the left in a unit of time. The transition probabilities for such a random walk from a point x is-an-element-of Z are determined by the vector P(x) is-an-element-of R(R+L+1). It is assumed that the sequence {p(x), x is-an-element-of Z) 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.