## 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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### References:

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