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May, 1985 Characterization and Domains of Attraction of $p$-Stable Random Compact Sets
Evarist Gine, Marjorie G. Hahn
Ann. Probab. 13(2): 447-468 (May, 1985). DOI: 10.1214/aop/1176993002


Let $(\mathscr{K} (\mathbb{B}), \delta)$ denote the nonempty compact subsets of a separable Banach space $\mathbb{B}$ topologized by the Hausdorff metric. Let $K, K_1, K_2$ be i.i.d. random compact convex sets in $\mathbb{B}. K$ is called $p$-stable if for each $\alpha, \beta \geq 0$ there exist compact convex sets $C$ and $D$ such that $\mathscr{L}(\alpha K_1 + \beta K_2 + C) = \mathscr{L}((\alpha^p + \beta^p)^{1/p}K + D)$ where + refers to Minkowski sum. A characterization of the support function for a compact convex set is provided and then utilized to determine all $p$-stable random compact convex sets. If $1 \leq p \leq 2$, they are trivial, merely translates of a fixed compact convex set by a $p$-stable $\mathbb{B}$-valued random variable. For $0 < p < 1$, they are translates of stochastic integrals with respect to nonnegative independently scattered $p$-stable measures on the unit ball of $\operatorname{co} \mathscr{K}(\mathbb{B})$. Deconvexification is also discussed. The domains of attraction of $p$-stable random compact convex sets with $0 < p < 1$ are completely characterized. The case $1 < p \leq 2$ is considered in Gine, Hahn and Zinn (1983). Precedents: Lyashenko (1983) and Vitale (1983) characterize the Gaussian random compact sets in $\mathbb{R}^d$.


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Evarist Gine. Marjorie G. Hahn. "Characterization and Domains of Attraction of $p$-Stable Random Compact Sets." Ann. Probab. 13 (2) 447 - 468, May, 1985.


Published: May, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0575.60014
MathSciNet: MR781416
Digital Object Identifier: 10.1214/aop/1176993002

Primary: 60D05
Secondary: 60B12 , 60E07 , 60F05

Keywords: $p$-stable , central limit theorems , domains of attraction , Minkowski sums , random sets , stochastic integrals , support functions

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 2 • May, 1985
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