Two-sided bounds and leading term for rates of convergence in the multivariate central limit theorem

For a sequence of independent and identically distributed random vectors, upper and lower bounds are obtained for the discrepancy between the probability measure Pn, induced by their normalized sum, and the Normal measure [Phi]. The upper and lower bounds are of the same order of magnitude. These results may be derived by a "leading term" approach, in which a signed measure Qn is introduced as a first order approximation to Pn - [Phi]. The purpose of this paper is to investigate properties of the leading term.