Large deviations for the first hitting time of a distant lower level by a random walk in random environment

We prove local theorems on large deviations for the moment T − n of hitting the level − n , n ∈ ℕ, by a random walk in a random environment.We obtain precise asymptotics of the probabilities of large deviations P( T −n = k ) which are uniform with respect to k ∈ ℕ such that ( n − k ) is even and k / n = k ( n )/ n lies in some compact set.