Convex and star-shaped sets associated with multivariate stable distributions. I: Moments and densities. (English) Zbl 1196.60029
Summary: It is known that each symmetric stable distribution in \(\mathbb R^d\) is related to a norm on \(\mathbb R^d\) that makes \(\mathbb R^d\) embeddable in \(L_p([0,1]\)). In the case of a multivariate Cauchy distribution the unit ball in this norm is the polar set to a convex set in \(\mathbb R^d\) called a zonoid. This work interprets symmetric stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate symmetric stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations.
MSC:
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60D05 | Geometric probability and stochastic geometry |
| 52A21 | Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) |
| 52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |
Keywords:
convex body; generalised function; Fourier transform; multivariate stable distribution; star body; spectral measure; support function; zonoidReferences:
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