A Lyapunov-type bound in Rd

Let X-1,... X-n be independent random vectors taking values in R-d such that EXk = 0 for all k. Write S = X-1 + ... + X-n. Assume that the covariance operator, say C-2, of S is invertible. Let Z be a centered Gaussian random vector such that covariances of S and Z are equal. Let C stand for the class of all convex subsets of R-d. We prove a Lyapunov-type bound for Delta = sup(A is an element of C) vertical bar P{S is an element of A} - P{Z is an element of A}vertical bar. Namely, Delta <= cd(1/4)beta with beta = beta(1) +...+ beta(1) and beta(k) = E vertical bar C-1Xk vertical bar(3), where c is an absolute constant. If the random variables X-1,..., X-n are independent and identically distributed and X-k has identity covariance, then the bound specifies to Delta <= cd(1/4)E vertical bar X-1 vertical bar(3)/root n. Whether one can remove the factor d(1/4) or replace it with a better one (eventually by 1) remains an open question.