Zooming in on a L\'evy process at its supremum

Let $M$ and $\tau$ be the supremum and its time of a L\'evy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $((X_{\tau+t\varepsilon}-M)/a_\varepsilon)_{t\in\mathbb R}$ as $\varepsilon\downarrow 0$, results in $(\xi_t)_{t\in\mathbb R}$ constructed from two independent processes having the laws of some self-similar L\'evy process $\widehat X$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $\widehat X$ under the zooming-in procedure as opposed to the classical zooming out of Lamperti (1962). As an application of this result we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result of Asmussen, Glynn and Pitman (1995) for the Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a L\'evy process at 0 is provided.