ON THE TIMES OF ATTAINING HIGH LEVELS BY A RANDOM WALK IN A RANDOM ENVIRONMENT

摘要

Let (p_i, q_i), i ∈ Z, be a sequence of independent identically distributed random vectors such that p_i, q_i > 0 and p_i +q_i = 1 a.s. for i ∈ Z. We consider a random walk in the random environ_ment {(p_i, q_i), i ∈ Z}. It is assumed that Eln(p0/q0) = 0 and 0 < Eln~2(q0/p0) < +∞. We study the times of attaining T_(n1) , . . . , T_(nm) of increasing levels n_1, . . . , n_m of order n. It is proved that the underlying probability space can be partitioned into random events (depending on n) such that their probabilities for large n are close to positive numbers, and on each such event, the set of times T_(n1) , . . . , T_(nm) is partitioned into consecutive groups such that elements of each group have the same order and are negligible compared with those of the successive group.