On convex hull and winding number of self-similar processes

Yu. Davydov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 154-162

Résumé

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.