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Bounds for stable measures of convex shells and stable approximations. (English) Zbl 1023.60024

Summary: The standard normal distribution \(\Phi\) on \(\mathbb{R}^d\) satisfies \(\Phi((\partial C)^\varepsilon)\leq c_d\varepsilon\), for all \(\varepsilon>0\) and for all convex subsets \(C\subset\mathbb{R}^d\), with a constant \(c_d \) which depends on the dimension \(d\) only. Here \(\partial C\) denotes the boundary of \(C\), and \((\partial C)^\varepsilon\) stands for the \(\varepsilon\)-neighborhood of \(\partial C\). Such bounds for the normal measure of convex shells are extensively used to estimate the accuracy of normal approximations. We extend the inequality to all (nondegenerate) stable distributions on \(\mathbb{R}^d\), with a constant which depends on the dimension, the characteristic exponent and the spectral measure of the distribution only. As a corollary we provide an explicit bound for the accuracy of stable approximations on the class of all convex subsets of \(\mathbb{R}^d\).

MSC:

60E07 Infinitely divisible distributions; stable distributions
60F05 Central limit and other weak theorems
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