A Nonsymmetric Correlation Inequality for Gaussian Measure

Let[mu]be a Gaussian measure (say, onRn) and letK,L[subset, double equals]Rnbe such thatKis convex,Lis a "layer" (i.e.,L={x: a[less-than-or-equals, slant]<x, u>[less-than-or-equals, slant]b} for somea, b[set membership, variant]Randu[set membership, variant]Rn), and the centers of mass (with respect to[mu]) ofKandLcoincide. Then[mu](K[intersection]L)[greater-or-equal, slanted][mu](K)·[mu](L). This is motivated by the well-known "positive correlation conjecture" for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimate[Phi](x)> 1-((8/[pi])1/2/(3x+(x2+8)1/2))e-x2/2,x>-1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu.