On importance sampling with mixtures for random walks with heavy tails

TL;DR

This paper investigates importance sampling methods for heavy-tailed random walks, demonstrating conditions for bounded and asymptotically optimal relative error, with practical examples and numerical evaluations.

Contribution

It introduces mixture-based importance sampling algorithms with theoretical guarantees for heavy-tailed random walks, including bounds and asymptotic optimality.

Findings

Mixture algorithms can achieve bounded relative error.

Asymptotic optimality of the proposed algorithms is established.

Numerical evaluations confirm theoretical results.

Abstract

Importance sampling algorithms for heavy-tailed random walks are considered. Using a specification with algorithms based on mixtures of the original distribution with some other distribution, sufficient conditions for obtaining bounded relative error are presented. It is proved that mixture algorithms of this kind can achieve asymptotically optimal relative error. Some examples of mixture algorithms are presented, including mixture algorithms using a scaling of the original distribution, and the bounds of the relative errors are calculated. The algorithms are evaluated numerically in a simple setting.