Concentration Inequalities for Multivariate Distributions: I. Multivariate Norivial Distributions

Let X rv Np(O, E), the p-variate normal distribution with mean °and positive definite covariance matrix .E. Anderson (1955) showed that if .E2 .E1 is positive semidefinite then PEl (C) 2: P E2(C) for every centrally symmetric (-C = convex set C ~ RP. Fefferman, Jodeit, and Perlman (1972) extended this result to elliptically contoured distributions. In the present paper similar multivariate f f" "f' . +' +..J r . /'1,1, " ~ conceturaiion ineqiuuuies are mvesugateo lor convex sets v tnat satisry a more general symmetry condition, namely invariance under a group G of orthogonal transformations on RP, 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 conclassical efficiency comparisons, expressed in terms of covariance <UL""L.CvvU, Minnesota, Mmneapolis, Minnesota, 55455. Research supported