Stein's method and a quantitative Lindeberg CLT for the Fourier transforms of random vectors

We use a multivariate version of Stein's method to establish a quantitative Lindeberg CLT for the Fourier transforms of random N-vectors. We achieve this, conceptually mainly by constructing a natural approach structure on N-random vectors overlying the topology of weak convergence, and technically mainly by deducing a specific integral representation for the Hessian matrix of a solution to the Stein equation with test function e(t)(x) = exp (-i Sigma(N)(k=1) t(k)x(k)), where t, x is an element of R-N. (C) 2015 Elsevier Inc. All rights reserved.