The Non-Singularity of Generalized Sample Covariance Matrices

Abstract

Let $X = (X_1, \\cdots, X_n)$ where the $X_i: p \\times 1$ are independent random vectors, and let $A: n \\times n$ be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix $XAX'$ be positive definite w.p.1. The results are applied to cases where $A$ has a particular form or $X_1, \\cdots, X_n$ are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix $\\sigma(X_i - \\bar{X})(X_i - \\bar{X})'$ is positive definite w.p. 1 $\\operatorname{iff} P\\lbrack X_1 \\in F\\rbrack = 0$ for every proper flat $F \\subset R^p$.