Estimates of the convergence rate in the “interval” CLT for sums of independent random vectors

Abstract

The remainder term in the multidimensional central limit theorem for sum of independent random vectors is evaluated taking into account the measure of the set which this sum falls into. The obtained estimates refine and supplement some previously known results due to the weakening of the moment conditions and a more accurate account of the dependence on the covariance matrix. A variant of the smoothing lemma, which in the one-dimensional case is an interval version of the known Esseen-Petrov lemma is also given. Refs 8.