Multivariate variational inequalities and the central limit theorem (Q1817516)

It is considered a continuous random vector with density \(f\). The components of this vector are uncorrelated with zero mean vector. The upper bound for the total variation distance between the d.f. of this vector and the multivariate standard normal d.f. is formulated using \(w\)-functions. These functions characterize the corresponding density \(f\). The total variation distance between the d.f. of the sum of \(n\) certain i.i.d. random vectors divided by \(\sqrt n\) and the multivariate standard normal d.f. converges to 0 as \(n\to\infty\), if the variance of the \(i\)th component of the \(w\)-function of the considered \(p\)-dimensional random vector exists for all \(i=1,\dots,p\). Variational inequalities in terms of a Fisher-type information matrix are given. The assumptions of Mayer-Wolf lead to the strengthened multivariate central limit theorem. The significant simplification of Mayer-Wolf's result is given here.