Asymptotics for Exponential Functionals of Random Walks

TL;DR

This paper analyzes the asymptotic behavior of exponential functionals of random walks, revealing how tail properties and path characteristics influence convergence rates and limiting coefficients.

Contribution

It provides a comprehensive asymptotic analysis, including convergence rates and explicit formulas for limiting coefficients based on path features.

Findings

Asymptotics depend on local minima and final values of the walk.

Convergence rates are characterized based on tail properties.

Explicit expressions for limiting coefficients are derived.

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 sample paths with either slowly decreasing local minimum or final value below a low level. Also, our thoughtful analysis of the interrelationship between the local minimum and the final value provides the exact expression for the limiting coefficients in terms of some transformations of the random walk.