The Smallest Eigenvalue of a Large Dimensional Wishart Matrix

Silverstein, Jack W.

Ann. Probab., Tome 13 (1985) no. 4, p. 1364-1368 / Harvested from Project Euclid

Résumé

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 $s \rightarrow \infty$.

Détail

Publié le : 1985-11-14
Classification:  Smallest eigenvalue of random matrix,  Gersgorin's theorem,  $\chi^2$ distribution,  60F15,  62H99