On the convex hull of symmetric stable processes

Kampf, Jürgen; Last, Günter; Molchanov, Ilya

doi:10.1090/S0002-9939-2012-11128-1

On the convex hull of symmetric stable processes
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by Jürgen Kampf, Günter Last and Ilya Molchanov PDF

Proc. Amer. Math. Soc. 140 (2012), 2527-2535 Request permission

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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