Geodesic
On convex hull and winding number of self-similar processes
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 154-162 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is well known that for a standard Brownian motion (BM) $\{B(t),\ t\geq0\}$ with values in $\mathbf R^d$, its convex hull $V(t)=\mathrm{conv}\{B(s),\ s\leq t\}$ with probability $1$ for each $t>0$ contains $0$ as an interior point (see Evans [3]). We also know that the winding number of a typical path of a two-dimensional BM is equal to $+\infty$. The aim of this article is to show that these properties aren't specifically “Brownian”, but hold for a much larger class of $d$-dimensional self-similar processes. This class contains in particular $d$-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Lévy processes. 
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Yu. Davydov. On convex hull and winding number of self-similar processes. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 154-162. http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a9/

[1] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic Theory, Springer, Berlin–Heidelberg, 1982 | MR | Zbl

[2] Yu. Davydov, “On convex hull of $d$-dimensional fractional Brownian motion”, Statist. Probab. Letters, 82:1 (2012), 37–39 | DOI | MR | Zbl

[3] S. N. Evans, “On the Hausdorff dimension of Brownian cone points”, Math. Proc. Cambridge Philos. Soc., 98 (1985), 343–353 | DOI | MR | Zbl

[4] F. Lavancier, A. Philippe, D. Surgailis, Covariance function of vector self-similar process, 2009, arXiv: 0906.4541v2 | MR

[5] R. Mansuy, M. Yor, Aspects of Brownian Motion, Springer, Berlin–Heidelberg, 2008 | MR | Zbl

[6] A. Račkauskas, Ch. Suquet, “Operator fractional Brownian motion as limit of polygonal lines processes in Hilbert space”, Stochast. Dynam., 11:1 (2011), 1–22 | DOI | MR

[7] G. Samorodnitsky, M. S. Taqqu, Stable non-Gaussian Random Processes, Chapman and Hall, New York, 1994 | MR | Zbl

[8] Y. Xiao, “Recent developments on fractal properties of Gaussian random fields”, Further Developments in Fractals and Related Fields, eds. J. Barral, S. Seuret, Springer, New York, 2013, 255–288 | DOI | MR | Zbl

[9] Y. Xiao, “Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields”, Probab. Theory Relat. Fields, 109 (1997), 129–157 | DOI | MR | Zbl

[10] R. C. Dalang, C. Mueller, Y. Xiao, Polarity of points for Gaussian random fields, 2015, arXiv: 1505.05417v1