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Zapiski Nauchnykh Seminarov POMI, 2015, Volume 441, Pages 154–162 (Mi znsl6231)  

On convex hull and winding number of self-similar processes

Yu. Davydov

University Lille 1, CNRS, UMR 8524, Laboratory P. Painlevé, France
References:
Abstract: 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.
Key words and phrases: Brownian motion, multi-dimensional fractional Brownian motion, stable Lévy processes, convex hull, winding number.
Received: 30.10.2015
English version:
Journal of Mathematical Sciences (New York), 2016, Volume 219, Issue 5, Pages 707–713
DOI: https://doi.org/10.1007/s10958-016-3140-3
Bibliographic databases:
Document Type: Article
UDC: 519
Language: English
Citation: Yu. Davydov, “On convex hull and winding number of self-similar processes”, Probability and statistics. Part 22, Zap. Nauchn. Sem. POMI, 441, POMI, St. Petersburg, 2015, 154–162; J. Math. Sci. (N. Y.), 219:5 (2016), 707–713
Citation in format AMSBIB
\Bibitem{Dav15}
\by Yu.~Davydov
\paper On convex hull and winding number of self-similar processes
\inbook Probability and statistics. Part~22
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 441
\pages 154--162
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6231}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3504503}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 219
\issue 5
\pages 707--713
\crossref{https://doi.org/10.1007/s10958-016-3140-3}
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  • https://www.mathnet.ru/eng/znsl/v441/p154
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