Lévy–LePage Series Representation of Stable Vectors: Convergence in Variation

Multidimensional stable laws Gα admit a well-known Lévy–LePage series representation\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$G_\alpha = \mathcal{L}\sum\limits_{j = 1}^\infty {\Gamma _j^{ - 1/\alpha } X_j } ,{\text{ 0 < }}\alpha {\text{ < 2}}$$ \end{document}where Γ1, Γ2,... are the successive times of jumps of a standard Poisson process, and X1, X2,... denote i.i.d. random vectors, independent of Γ1, Γ2,.... We present (asymptotically) optimal bounds for the total variation distance between a stable law and the distribution of a partial sum of the Lévy–LePage series. In the one-dimensional case similar results were obtained earlier by Bentkus, Götze, and Paulauskas.