The Annals of Probability

Characterization and Domains of Attraction of $p$-Stable Random Compact Sets

Evarist Gine and Marjorie G. Hahn

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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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Ann. Probab., Volume 13, Number 2 (1985), 447-468.

First available in Project Euclid: 19 April 2007

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Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Random sets domains of attraction $p$-stable support functions stochastic integrals central limit theorems Minkowski sums


Gine, Evarist; Hahn, Marjorie G. Characterization and Domains of Attraction of $p$-Stable Random Compact Sets. Ann. Probab. 13 (1985), no. 2, 447--468. doi:10.1214/aop/1176993002.

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