We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance covariance matrix when the dimensions of the vectors and the sample size tend to infinity simultaneously. This limit is valid when the vectors are independent or weakly dependent through a finite-rank perturbation. It is also universal and independent of the details of the distributions of the vectors. Furthermore, the top eigenvalues of this distance covariance matrix are shown to obey an exact phase transition when the dependence of the vectors is of finite rank. This finding enables the construction of a new detector for such weak dependence where classical methods based on large sample covariance matrices or sample canonical correlations may fail in the considered high-dimensional framework.
翻译:在本文中,我们引入了一个新的随机矩阵模型,称为“距离共变矩阵”,其归正追踪相当于距离共变。当矢量的尺寸和样本大小倾向于不宽时,我们首先为距离共变矩阵的双值分布得出一个确定性限值。当矢量的尺寸和样本大小同时不宽时,这一限值是有效的,当矢量独立时,或者通过有限级别的扰动而弱于依赖性时,这一限值也是有效的。它也是普遍性的,独立于矢量分布的细节。此外,在矢量依赖性为定级时,这种距离共变异矩阵的顶值显示符合一个精确的阶段过渡。这一发现使得能够为这种微弱依赖性而建造一个新的探测器,因为基于大样本共变基质矩阵或样本可感应关系的传统方法在考虑的高维框架中可能失败。