Zooming in on a Lévy process at its supremum

Let M and . be the supremum and its time of a Lévy process X on some finite time interval. It is shown that zooming in on X at its supremum, that is, considering ((X.+t..M)/a.)t.R as ..0, results in (.t)t.R constructed from two independent processes having the laws of some self-similar Lévy process .X conditioned to stay positive and negative. This holds when X is in the domain of attraction of .X under the zooming-in procedure as opposed to the classical zooming out [Trans. Amer. Math. Soc. 104 (1962) 62.78]. 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 in [Ann. Appl. Probab. 5 (1995) 875.896] for a linear Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process is provided.