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Convex hulls of Lévy processes. (English) Zbl 1348.60071

Summary: Let \(X(t)\), \(t\geq 0\), be a Lévy process in \(\mathbb{R} ^d\) starting at the origin. We study the closed convex hull \(Z_s\) of \(\{X(t): 0\leq t\leq s\}\). In particular, we provide conditions for the integrability of the intrinsic volumes of the random set \(Z_s\) and find explicit expressions for their means in the case of symmetric \(\alpha \)-stable Lévy processes. If the process is symmetric and each of its one-dimensional projections is non-atomic, we establish that the origin a.s. belongs to the interior of \(Z_s\) for all \(s>0\). Limit theorems for the convex hull of Lévy processes with normal and stable limits are also obtained.

MSC:

60G51 Processes with independent increments; Lévy processes
60D05 Geometric probability and stochastic geometry
60F99 Limit theorems in probability theory
52A22 Random convex sets and integral geometry (aspects of convex geometry)