Product Inequalities Involving the Multivariate Normal Distribution

Suppose Y′ = (Y′ 1, …, Y′ k ) possesses a multivariate normal distribution with mean vector 0 and positive semidefinite covariance matrix Σ. If Ci ⊂ Rpi denote convex regions symmetric about the origin, then conditions are given such that and/or obtain. These conditions imply that chi-squared random variables defined from a multivariate normal distribution are always positively dependent and nonnegatively correlated. Other applications involve conservative simultaneous confidence regions in a multivariate regression setting.