A note on Veraverbeke's theorem

TL;DR

This paper provides an elementary probabilistic proof of Veraverbeke's Theorem, elucidating how the maximum of a heavy-tailed, negatively drifted random walk is typically achieved by a single large jump.

Contribution

It offers a simplified proof of Veraverbeke's Theorem, enhancing understanding of the maximum's asymptotic distribution in heavy-tailed random walks.

Findings

Maximum is attained through a single large jump

Elementary proof clarifies the theorem's principles

Deepens insight into heavy-tailed random walk behavior

Abstract

We give an elementary probabilistic proof of Veraverbeke's Theorem for the asymptotic distribution of the maximum of a random walk with negative drift and heavy-tailed increments. The proof gives insight into the principle that the maximum is in general attained through a single large jump.