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• W2086839512 abstract "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 $mathbb{R}^d$." @default.
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• W2086839512 date "1985-05-01" @default.
• W2086839512 modified "2023-09-26" @default.
• W2086839512 title "Characterization and Domains of Attraction of $p$-Stable Random Compact Sets" @default.
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• W2086839512 doi "https://doi.org/10.1214/aop/1176993002" @default.
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