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 $\{Y_n^{(a)}:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X_t^{(a)}:t\ge 0\}$ be a Lévy processes such that $X_1^{(a)}\stackrel{d}{=}{Y}_1^{(a)}$, $\mathbb{E}{X}_1^{(a)}<0$ and $\mathbb{E}{X}_1^{(a)}\uparrow 0$ as $a\downarrow 0$. Let $S_n^{(a)}={\sum }_{k=1}^n{Y}_k^{(a)}$. Then, under some mild assumptions, $\hat{I}”(a){\max }_{n\ge 0}{S}_n^{(a)}\stackrel{d}{\to }R⟺\hat{I}”(a){\sup }_{t\ge 0}{X}_t^{(a)}\stackrel{d}{\to }R$, for some random variable $R$ and some function $\hat{I}”(\cdot )$. We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.