Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotics for Exponential Functionals of Random Walks

Wei Xu

TL;DR

This work characterizes the asymptotics of E[F(I_n)] for the exponential functional I_n = ∑_{k=1}^n e^{-S_k} of a one-dimensional random walk with general light- or heavy-tailed steps. By combining fluctuation theory, Esscher transforms, and a careful pathwise decomposition around local minima and large early jumps, the authors derive six distinct regimes determined by the infimum of the Laplace transform of X and the drift under the transformed measure, each with explicit limiting coefficients. The main contributions are (i) precise decay rates involving exponential factors times regularly varying modifiers, and (ii) explicit representations of the limiting constants in terms of transformed random-walk functionals and ladder-Height renewal structures. These results advance understanding of exponential functionals in random environments and offer a blueprint for extending to Lévy-process functionals and related stochastic models.

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.

Asymptotics for Exponential Functionals of Random Walks

TL;DR

This work characterizes the asymptotics of E[F(I_n)] for the exponential functional I_n = ∑_{k=1}^n e^{-S_k} of a one-dimensional random walk with general light- or heavy-tailed steps. By combining fluctuation theory, Esscher transforms, and a careful pathwise decomposition around local minima and large early jumps, the authors derive six distinct regimes determined by the infimum of the Laplace transform of X and the drift under the transformed measure, each with explicit limiting coefficients. The main contributions are (i) precise decay rates involving exponential factors times regularly varying modifiers, and (ii) explicit representations of the limiting constants in terms of transformed random-walk functionals and ladder-Height renewal structures. These results advance understanding of exponential functionals in random environments and offer a blueprint for extending to Lévy-process functionals and related stochastic models.

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.
Paper Structure (17 sections, 45 theorems, 250 equations, 1 figure)

This paper contains 17 sections, 45 theorems, 250 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Random walks
  4. Main results
  5. The oscillating cases
  6. The negative drift cases
  7. Auxiliary results for random walks
  8. Oscillating random walk
  9. Random walk with negative drift
  10. Proof for Theorems \ref{['MainThm.01']}-\ref{['MainThm.03']}
  11. Proof for Theorem \ref{['MainThm.01']}
  12. Proof for Theorem \ref{['MainThm.02']}
  13. Proof for Theorem \ref{['MainThm.03']}
  14. Proof for Theorems \ref{['MainThm.06']}-\ref{['MainThm.05']}
  15. Proof for Theorem \ref{['MainThm.06']}
  16. ...and 2 more sections

Key Result

Lemma 2.4

The following assertions are equivalent: (i) $I_\infty<\infty$ a.s.; (ii) $\mathbf{P}( I_\infty<\infty )>0$; (iii) $S$ drifts to $\infty$, i.e., $S_n\to \infty$ a.s. as $n\to\infty$.

Figures (1)

  • Figure 1: Figures (a)-(c) draw the three possibilities of the Laplace transform of random walk $S$ oscillating under $\mathbf{P}^{(\Lambda)}$, in which $\mathcal{D}^+_{\mathcal{L}_X} \supset [0,\Lambda]$. Figures (d)-(f) draw the three possibilities of the Laplace transform of random walk $S$ with negative drift under $\mathbf{P}^{(\Lambda)}$, in which $\mathcal{D}^+_{\mathcal{L}_X}=[0,\Lambda]$. The thick dotted lines in each figure represent the set besides $[0,\Lambda]$ on which the Laplace transform may be finite.

Theorems & Definitions (47)

  • Lemma 2.4
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.10
  • Theorem 2.11
  • Theorem 2.12
  • Theorem 2.15
  • Theorem 2.17
  • Remark 3.1
  • Lemma 3.2
  • ...and 37 more