## Determination of the Limiting Coefficient for Exponential Functionals of Random Walks with Positive Drift

J. Math. Sci. Univ. Tokyo
Vol. 5 (1998), No. 2, Page 299--332.

Hirano, Katsuhiro
Determination of the Limiting Coefficient for Exponential Functionals of Random Walks with Positive Drift
Let $(S_n, n\ge 1)$ be a random walk satisfying $ES_1>0$ and $h$ be a Laplace transform of a non-negative finite measure on $(0, \infty)$. Under additional conditions of $S_1$ and $h$, we consider the asymptotic behavior of $Eh(\sum_{i=1}^ne^{S_i})$. 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 conditions. This solution corresponds to the Green function of $2^{-1}e^{-x}\triangle$ on {\bf R}. We apply our result to random processes in random media. Moreover we obtain the random walk analogue of Kotani's limit theorem for Brownian motion.