Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Sz\'{e}kely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric L\'evy measures and the corresponding real-valued continuous negative definite functions. The L\'evy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework. From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance multivariance: New dependence measures for random vectors] to the present article.
翻译:距离共变量是测量两个随机矢量依赖性的一个数量。 我们显示, Sz\ { { e}kely、 Rizzo 和 Bakirov 所引入和开发的原始概念可以嵌入一个基于对称L\ evy 测量法和相应的实际价值持续负确定函数的更一般性框架。 L\' evy 测量法取代了距离共变量原始定义中所使用的重量函数。 远程共变量的所有基本特性都保存在这个新框架中。 从实际的角度看, 允许对潜在随机变量的不那么严格的时刻条件, 并且可以使用比 Euclidean 距离( 如 Minkowski 距离) 以外的其他距离函数。 最重要的是, 它可以作为距离多变量的基本建筑块, 用来测量和估计多个随机矢量的依赖性, 这是在本文的后续文件[ Dustance 多变量: 中引入的任意矢量的新依赖性度度度度度度度度度度度度度度度度度度度度度度度度度度度度度度。