On the Euclidean distance between two Gaussian points and the normal covariogram of \(\mathbb{R}^d\) (Q6602405)

The covariogram of a bounded body \(\mathbb D\subset \mathbb R^d\), defined at a point \(t\in \mathbb R^d\) as the Lebesgue measure of the intersection of \(\mathbb D\) and the translated copy of \(\mathbb D\) by the vector \(t\), has a natural connection to the Euclidean distance of two uniform random points in \(\mathbb D\). In this paper, the authors consider the generalisation of this interpoint distance to \(\mathbb R^d\) by examining the Euclidean distance of two Gaussian points in \(\mathbb R^d\), and they extend the concept of the covariogram from a bounded body to the entire space \(\mathbb R^d\) with the use of this random variable. Representations of the distribution and density function of the distance are obtained, as well as precise bounds for its moments.