Bounds for stable measures of convex shells and stable approximations
成果类型：
Article
署名作者：
Bentkus, V; Juozulynas, A; Paulauskas, V
署名单位：
Vilnius University; Vilnius University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
2000
页码：
1280-1300
关键词：
摘要：
The standard normal distribution Phi on R-d satisfies Phi ((partial derivativeC)(epsilon)) less than or equal to c(d)epsilon, for all epsilon > 0 and for all convex subsets C subset of R-d, with a constant c(d) which depends on the dimension d only. Here partial derivativeC denotes the boundary of C, and (partial derivativeC)(epsilon) stands for the epsilon -neighborhood of partial derivativeC. 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 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 R-d.