Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime

TL;DR

This paper develops a technique to derive limit theorems for the supremum of Lévy processes in heavy-traffic conditions, linking them to their random walk counterparts, and presents new results in this regime.

Contribution

It introduces a method to obtain limit theorems for Lévy process suprema from random walk limits, advancing understanding in heavy-traffic analysis.

Findings

Established a convergence equivalence between scaled suprema of Lévy processes and random walks.

Derived new limit theorems for Lévy process suprema in heavy-traffic regimes.

Provided mild assumptions under which the convergence results hold.

Abstract

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_t:t\ge 0\}$ be a L\'evy processes such that $X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}$, $\mathbb E X_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$ as $a\downarrow0$. Let $S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime.