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 et(x)=exp⁡(−i∑k=1Ntkxk), where t,x∈RNt,x∈RN.