Determination of the limiting coefficient for exponentials functional of random walks with positive drift

Abstract. Let (Sn, n ≥ 1) be a random walk satisfying ES1> 0 and h be a Laplace transform of a non-negative finite measure on (0,∞). Under additional conditions of S1 and h, we consider the asymptotic behavior of Eh( ∑n i=1 e Si). In particular we determine the limiting coefficient for asymptotic of this quantity in terms of the unique solution of the certain functional equation with boundary con-ditions. This solution corresponds to the Green function of 2−1e−x on R. We apply our result to random processes in random media. Moreover we obtain the random walk analogue of Kotani’s limit the-orem for Brownian motion