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

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 $\mathbb{R}^d$.