Molchanov, Ilya; Wespi, Florian Convex hulls of Lévy processes. (English) Zbl 1348.60071 Electron. Commun. Probab. 21, Paper No. 69, 11 p. (2016). 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. Cited in 6 Documents 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) Keywords:Lévy processes; convex hull; intrinsic volume; mixed volume; stable law; limit theorems × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid