Electronic Communications in Probability

Convex hulls of Lévy processes

Ilya Molchanov and Florian Wespi

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Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 69, 11 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1475266872

Digital Object Identifier
doi:10.1214/16-ECP19

Mathematical Reviews number (MathSciNet)
MR3564216

Zentralblatt MATH identifier
1348.60071

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Keywords
convex hull intrinsic volume mixed volume Lévy process stable law

Rights
Creative Commons Attribution 4.0 International License.

Citation

Molchanov, Ilya; Wespi, Florian. Convex hulls of Lévy processes. Electron. Commun. Probab. 21 (2016), paper no. 69, 11 pp. doi:10.1214/16-ECP19. https://projecteuclid.org/euclid.ecp/1475266872


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