## The Annals of Statistics

### 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 $F \subset R^p$.

#### Article information

Source
Ann. Statist., Volume 1, Number 4 (1973), 710-717.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176342465

Digital Object Identifier
doi:10.1214/aos/1176342465

Mathematical Reviews number (MathSciNet)
MR341745

Zentralblatt MATH identifier
0261.62037

JSTOR