Bounds for stable measures of convex shells and stable approximations

Abstract

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 $\mathbb{R}^d$ .