High-Dimensional Mahalanobis Distances of Complex Random Vectors

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio $c=p/n\to \infty$. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix $\mathit{F}={\mathit{S}}_2^{-1}{\mathit{S}}_1$—the product of a sample covariance matrix ${\mathit{S}}_1$ (from the independent variable array $(\mathit{b}e{\left(Z_i\right)}_{1\times n}$) with the inverse of another covariance matrix ${\mathit{S}}_2$ (from the independent variable array ${\left({\mathit{Z}}_{j\ne i}\right)}_{p\times n}$)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components ${\mathit{S}}_1$ and ${\mathit{S}}_2$ of the F-matrix is not required.