Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

Abstract

In this paper we derive a technique of obtaining limit theorems for suprema of Lévy 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 Lévy 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, $Δ(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iffΔ(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $Δ(\cdot)$. We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.