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July 2000 Bounds for stable measures of convex shells and stable approximations
V. Bentkus, A. Juozulynas, V. Paulauskas
Ann. Probab. 28(3): 1280-1300 (July 2000). DOI: 10.1214/aop/1019160335


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 $(\partial C)^\epsilon$ stands for the $\epsilon$-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 $\mathbb{R}^d$ .


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V. Bentkus. A. Juozulynas. V. Paulauskas. "Bounds for stable measures of convex shells and stable approximations." Ann. Probab. 28 (3) 1280 - 1300, July 2000.


Published: July 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1023.60024
MathSciNet: MR1797313
Digital Object Identifier: 10.1214/aop/1019160335

Primary: 60E07
Secondary: 60F05

Keywords: $\varepsilon$-strip , Convergence rates , convex set , convex shell , stable approximations , Stable measure

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 3 • July 2000
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