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 P\X₁ < 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 ᵢⱼ₈₉, \₈₉\ 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).