The Smallest Eigenvalue of a Large Dimensional Wishart Matrix

Abstract

For positive integers $s, n$ let $M_s = (1/s)V_sV^T_s$, where $V_s$ is an $n \times s$ matrix composed of i.i.d. $N(0, 1)$ random variables. Assume $n = n(s)$ and $n/s \rightarrow y \in (0, 1)$ as $s \rightarrow \infty$. Then it is shown that the smallest eigenvalue of $M_s$ converges almost surely to $(1 - \sqrt y)^2$ as $s \rightarrow \infty$.