Levy-LePage series representation of stable vectors: Convergence in variation

Multidimensional stable laws G(alpha) admit a well-known Levy-LePage series representation G(alpha)=L(Sigma (alpha)(j=1)Gamma X--1/alpha(j)j), 0 < alpha <2 where Gamma (1), Gamma (2), .... are the successive times of jumps of a standard Poisson process, and X-1, X-2,.... denote i.i.d. random vectors, independent of Gamma (1), Gamma (2),.... We present (asymptotically) optimal bounds for the total variation distance between a stable law and the distribution of a partial sum of the Levy-LePage series. In the one-dimensional case similar results were obtained earlier by Bentkus, Gotze, and Paulauskas.