On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations

For a random vector (X₁,⋯, Xₖ) having a k-variate normal distribution with zero mean values, Slepian [16] has proved that the probability P₁ < c₁,⋯, Xₖ < cₖ\ is a non-decreasing function of correlations. The present paper deals with the "two-sided" analogue of this problem, namely, if also the probability P\|X₁| < c₁,⋯, |Xₖ| < cₖ\ is a non-decreasing function of correlations. It is shown that this is true in the important special case where the correlations are of the form λᵢλⱼρᵢⱼ, \ρᵢⱼ\ being some fixed correlation matrix (Section 1), and that it is true locally in the case of equicorrelated variables (Section 3). However, some counterexamples are offered showing that a complete analogue of Slepian's result does not hold in general (Section 4). Some applications of the main positive result are mentioned briefly (Section 2).