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.

中文翻译：

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关于两个高斯点之间的欧氏距离和 $$\boldsymbol{\mathbb{R}}^{\boldsymbol{d}}$$ 的正态协方差图

摘要

通过获取分布和概率的积分表示，协变函数的概念从\(\mathbb{R}^{d}\)中的有界凸体扩展到整个空间\(\mathbb{R}^{d}\)两个d维高斯点之间的欧几里得距离的密度函数，这些点具有由协方差矩阵控制的相关坐标。当\(d=2\)时，获得密度函数的封闭式表达式。根据协方差矩阵的极值特征值找到所考虑距离矩的精确界限。