## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 39, Number 5 (1968), 1425-1434.

### On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations

#### Abstract

For a random vector $(X_1,\cdots, X_k)$ having a $k$-variate normal distribution with zero mean values, Slepian [16] has proved that the probability $P\{X_1 < c_1,\cdots, X_k < c_k\}$ is a non-decreasing function of correlations. The present paper deals with the "two-sided" analogue of this problem, namely, if also the probability $P\{|X_1| < c_1,\cdots, |X_k| < c_k\}$ 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 $\lambda_i\lambda_j\rho_{ij}, \{\rho_{ij}\}$ 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).

#### Article information

**Source**

Ann. Math. Statist., Volume 39, Number 5 (1968), 1425-1434.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177698122

**Digital Object Identifier**

doi:10.1214/aoms/1177698122

**Mathematical Reviews number (MathSciNet)**

MR230403

**Zentralblatt MATH identifier**

0169.50102

**JSTOR**

links.jstor.org

#### Citation

Sidak, Zbynek. On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations. Ann. Math. Statist. 39 (1968), no. 5, 1425--1434. doi:10.1214/aoms/1177698122. https://projecteuclid.org/euclid.aoms/1177698122