Efficient rare-event simulation for the maximum of heavy-tailed random walks

TL;DR

This paper introduces a state-dependent importance sampling algorithm for efficiently estimating the tail probability of the maximum of a random walk with i.i.d. increments, effective for both light and heavy tails, with proven asymptotic efficiency.

Contribution

It presents a novel, state-dependent importance sampling method that is strongly efficient for heavy and light-tailed distributions, extending previous algorithms and providing new analytical tools.

Findings

Algorithm is strongly efficient for heavy and light tails.

Asymptotic vanishing variance for heavy-tailed cases.

New Lyapunov inequalities aid in efficiency analysis.

Abstract

Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.