Asymptotics for Exponential Functionals of Random Walks

Abstract

This paper provides a detailed description for the asymptotics of exponential functionals of random walks with light/heavy tails. We give the convergence rate based on the key observation that the asymptotics depends on the simple paths with either slowly decreasing local minimum or final value below a low level. Also, our careful analysis of the interrelationship between the local minimum and the final value provides the exact expression of the limiting coefficient in terms of some transformations of the random walk.