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On the convex hull of symmetric stable processes

Authors: Jürgen Kampf, Günter Last and Ilya Molchanov
Journal: Proc. Amer. Math. Soc. 140 (2012), 2527-2535
MSC (2010): Primary 60G52; Secondary 28A75, 60D05
Published electronically: January 18, 2012
MathSciNet review: 2898714
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Abstract: Let $ \alpha \in (1,2]$ and $ X$ be an $ \mathbb{R}^d$-valued symmetric $ \alpha $-stable Lévy process starting at 0. We consider the closure $ S_t$ of the path described by $ X$ on the interval $ [0,t]$ and its convex hull $ Z_t$. The first result of this paper provides a formula for certain mean mixed volumes of $ Z_t$ and in particular for the expected first intrinsic volume of $ Z_t$. The second result deals with the asymptotics of the expected volume of the stable sausage $ Z_t+B$ (where $ B$ is an arbitrary convex body with interior points) as $ t\to 0$. For this we assume that $ X$ has independent components.

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Additional Information

Jürgen Kampf
Affiliation: AG Statistik, TU Kaiserslautern, 67653 Kaiserslautern, Germany

Günter Last
Affiliation: Institut für Stochastik, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

Ilya Molchanov
Affiliation: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland

Received by editor(s): December 9, 2010
Received by editor(s) in revised form: February 25, 2011
Published electronically: January 18, 2012
Additional Notes: The third author was partially supported by Swiss National Science Foundation Grant No. 200021-126503.
The authors are grateful to the referee for a careful reading of the paper.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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