The Multivariate Central Limit Theorem for Regular Convex Sets

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random vectors in $R_k$. Let $P_n$ denote the probability measure induced by the normalized sum and let $Q_n$ denote the multivariate Edgeworth signed measure with terms through $n^{-\frac{1}{2}}$. If $C$ is a member of a class of convex bodies whose boundaries are sufficiently smooth and possess positive Gaussian curvatures, and $X_1$ has fourth moments, it is shown that $P_n(C) - Q_n(C) = 0(n^{-k/(k + 1)})$ where the bound is uniform. If, moreover, $X_1$ has a nonlattice distribution, the difference is $o(n^{-k/(k + 1)})$.