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

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let <EquationSource Format="TEX">${Y^{(a)}_{n}:nge1}$</EquationSource> be a sequence of independent and identically distributed random variables and <EquationSource Format="TEX">${X^{(a)}_{t}:tge0}$</EquationSource> be a Lévy process such that <EquationSource Format="TEX">$X_{1}^{(a)}stackrel{d}{=}Y_{1}^{(a)}$</EquationSource> , <EquationSource Format="TEX">$mathbb{E}X_{1}^{(a)}<0$</EquationSource> and <EquationSource Format="TEX">$mathbb{E}X_{1}^{(a)}uparrow0$</EquationSource> as a↓0. Let <EquationSource Format="TEX">$S^{(a)}_{n}=sum _{k=1}^{n} Y^{(a)}_{k}$</EquationSource> . Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.