## Electronic Communications in Probability

### Convex hulls of Lévy processes

#### 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
Accepted: 26 August 2016
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.ecp/1475266872

Digital Object Identifier
doi:10.1214/16-ECP19

Mathematical Reviews number (MathSciNet)
MR3564216

Zentralblatt MATH identifier
1348.60071

#### 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

#### References

• [1] Jean Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996.
• [2] Ronen Eldan, Volumetric properties of the convex hull of an $n$-dimensional Brownian motion, Electron. J. Probab. 19 (2014), no. 45, 34.
• [3] Steven N. Evans, On the Hausdorff dimension of Brownian cone points, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 2, 343–353.
• [4] Shui Feng, The Poisson-Dirichlet Distribution and Related Topics, Probability and its Applications (New York), Springer, Heidelberg, 2010.
• [5] Z. Kabluchko and D. N. Zaporozhets, Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 408 (2012), no. Veroyatnost i Statistika. 18, 187–196, 327.
• [6] Jürgen Kampf, Günter Last, and Ilya Molchanov, On the convex hull of symmetric stable processes, Proc. Amer. Math. Soc. 140 (2012), no. 7, 2527–2535.
• [7] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, A new affine invariant for polytopes and Schneider’s projection problem, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1767–1779.
• [8] Satya N. Majumdar, Alain Comtet, and Julien Randon-Furling, Random convex hulls and extreme value statistics, J. Stat. Phys. 138 (2010), no. 6, 955–1009.
• [9] Mark M. Meerschaert and Hans-Peter Scheffler, Limit Distributions for Sums of Independent Random Vectors, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 2001.
• [10] Ilya Molchanov, Theory of Random Sets, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2005.
• [11] Ilya Molchanov, Convex and star-shaped sets associated with multivariate stable distributions. I. Moments and densities, J. Multivariate Anal. 100 (2009), no. 10, 2195–2213.
• [12] Marian Mureşan, A Concrete Approach to Classical Analysis, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2009.
• [13] William E. Pruitt, The growth of random walks and Lévy processes, Ann. Probab. 9 (1981), no. 6, 948–956.
• [14] Ken-iti Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 2005.
• [15] Rolf Schneider, Convex Bodies: the Brunn-Minkowski Theory, expanded ed., Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014.
• [16] Richard A. Vitale, Expected absolute random determinants and zonoids, Ann. Appl. Probab. 1 (1991), no. 2, 293–300.
• [17] Vladislav Vysotsky and Dmitry Zaporozhets, Convex hulls of multidimensional random walks, arXiv:1506.07827.
• [18] Andrew R. Wade and Chang Xu, Convex hulls of random walks and their scaling limits, Stochastic Process. Appl. 125 (2015), no. 11, 4300–4320.
• [19] Ward Whitt, Stochastic-Process Limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002.