# A Dichotomy for Sampling Barrier-Crossing Events of Random Walks with Regularly Varying Tails

## TL;DR

This paper investigates the sampling of random walk paths crossing a barrier with heavy-tailed step sizes, identifying when a specific change of measure enables exact simulation based on the tail index.

## Contribution

It characterizes the conditions under which Blanchet and Glynn's change of measure is suitable for exact sampling, based on the tail index of the step size distribution.

## Key findings

- Suitable change of measure exists if and only if tail index α is in (1, 3/2)
- The study links tail behavior to the feasibility of exact simulation methods
- Provides criteria for when a known measure can be used for barrier-crossing sampling

## Abstract

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index $\alpha$ of the right tail lies in the interval $(1, \, 3/2)$.