Open Access
2016 Convex hulls of Lévy processes
Ilya Molchanov, Florian Wespi
Electron. Commun. Probab. 21: 1-11 (2016). DOI: 10.1214/16-ECP19


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 its one-dimensional projection 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.


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Ilya Molchanov. Florian Wespi. "Convex hulls of Lévy processes." Electron. Commun. Probab. 21 1 - 11, 2016.


Received: 31 December 2015; Accepted: 26 August 2016; Published: 2016
First available in Project Euclid: 30 September 2016

zbMATH: 1348.60071
MathSciNet: MR3564216
Digital Object Identifier: 10.1214/16-ECP19

Primary: 52A22 , 60D05 , 60G51

Keywords: Convex hull , intrinsic volume , Lévy process , mixed volume , Stable law

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