On the Euclidean Distance between Two Gaussian Points and the Normal Covariogram of $$\boldsymbol{\mathbb{R}}^{\boldsymbol{d}}$$

Abstract

The concept of covariogram is extended from bounded convex bodies in \(\mathbb{R}^{d}\) to the entire space \(\mathbb{R}^{d}\) by obtaining integral representations for the distribution and probability density functions of the Euclidean distance between two \(d\)-dimensional Gaussian points that have correlated coordinates governed by a covariance matrix. When \(d=2\), a closed-form expression for the density function is obtained. Precise bounds for the moments of the considered distance are found in terms of the extreme eigenvalues of the covariance matrix.