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

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 $\{Y_n^{(a)}:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X_t^{(a)}:t\ge 0\}$ be a L'evy processes such that $X_1^{(a)}{Y}_1^{(a)}$, $\mathbb{E}{X}_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$as$a\downarrow0$\mathrm{.}Let$S^{(a)}n=\sum{k=1}^n Y^{(a)}k$. Then, under some mild assumptions, $\Delta(a)\max{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$,forsomerandomvariable$R$andsomefunction$\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L'evy processes in the heavy-traffic regime.