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\'{e}vy measures and the corresponding real-valued continuous negative definite functions. The L\'{e}vy 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 (submitted). Revised version of arXiv: 1711.07775v1] 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} keyly、 Rizzo 和 Bakirov 最初提出和开发的概念可以嵌入一个基于对称 L\ { e} 量和相应的实际价值连续负确定函数的更一般性框架。 L\ { { e} 量度取代了距离共变量原定义中所使用的重量函数。 距离共变量的所有基本特性都保存在这个新框架中。 从实际角度看, 允许对潜在随机变量采用较少限制的瞬间条件, 并且可以使用比 Euclidean 距离(例如 Minkowski 距离) 以外的其他距离函数。 最重要的是, 它作为距离多变量的基本建筑块, 是测量和估计多随机矢量依赖性的一个数量, 这是在后续文件中引入的[ distent 倍变量: 随机矢量的新依赖性措施( 已提交)。 修订后的 arXiv 版本: 171.0 7775v1) 到本条。