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 $\{Y_n^{(a)}:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X_t^{(a)}:t\ge 0\}$ be a Lévy process such that $X_1^{(a)}\stackrel{d}{=}{Y}_1^{(a)}$ , $𝔼X_1^{(a)}\text{<}0$ and $𝔼X_1^{(a)}\uparrow 0$ as a ↓ 0. Let $S_n^{(a)}={\sum }_{k=1}^n{Y}_k^{(a)}$ . Then, under some mild assumptions, Δ(a)maxn≥0Sn^(a(→^dℛ⇔ Δ(a)t≥0Xt^(a)→^dℛ, for some random variable ℛ 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.