  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.
