Concentration inequalities for multivariate distributions: I. multivariate normal distributions

Let X ~ Np(0, [Sigma]), the p-variate normal distribution with mean 0 and positive definite covariance matrix [Sigma]. Anderson (1955) showed that if [Sigma]2 - [Sigma]1 is positive semidefinite then P[Sigma]1(C) [greater-or-equal, slanted] P[Sigma]2(C) for every centrally symmetric (- C = C) convex set C[subset, double equals]p. Fefferman, Jodeit and Perlman (1972) extended this result to elliptically contoured distributions. In the present study similar multivariate concentration inequalities are investigated for convex sets C that satisfy a more general symmetry condition, namely invariance under a group G of orthogonal transformations on p, as well as for non-convex sets C that are monotonically decreasing with respect to a pre-ordering determined by G. Both new results and counterexamples are presented. Concentration inequalities may be used to convert classical efficiency comparisons, expressed in terms of covariance matrices, into comparisons of probabilities of multivariate regions.