ON THE TIME OF REACHING A HIGH LEVEL BY A TRANSIENT RANDOM WALK IN A RANDOM ENVIRONMENT

Let a sequence of independent identically distributed pairs of random variables (pi, qi), i is an element of Z, be given, with p(0) + q(0) = 1 and p(0) > 0, q(0) > 0 a.s. We consider a random walk in the random environment (p(i), q(i)), i is an element of Z. This means that under a fixed environment a walking particle located at some moment in a state i jumps either to the state (i - 1) with probability p(i) or to the state (i - 1) with probability q(i). It is assumed that E log(p(0)/q(0)) < 0, i.e., that the random walk tends with time to - infinity. The set of such random walks may be divided into three types according to the value of the quantity E ((p(0)/q(0)) log(p(0)/q(0))). In the case when the expectation above is zero we prove a limit theorem as n -> infinity for the of time distribution of reaching the level n by the mentioned random walk.