Nonuniform estimate of the rate of convergence to a stable law in the multidimensional central limit theorem
Let \(X_ 1,X_ 2,...,X_ n\in {\mathbb{R}}^ k\) be independent random vectors with the same distribution F. Let Y denote a spherically symmetric stable random vector with exponent \(0<\alpha \leq 2\) and distribution G. Then \[ | P(X_ 1+..+X_ n\in n^{1/\alpha}A)- G(A)| \leq k^{1+\alpha /2}c(\alpha)n^{1-1/\alpha}\int | x|^{1+\alpha}| F-G| (dx)\cdot (\inf_{x\in \partial A}| x|)^{-1-\alpha} \] for any convex \(A\subset {\mathbb{R}}^ k\). The constant c(\(\alpha\)) depends on \(\alpha\) only. The existence of such estimates is discussed in the more general case of a strictly stable vector Y.
scientific article; zbMATH DE number 4174021