The Smallest Eigenvalue of a Large Dimensional Wishart Matrix

Abstract

For positive integers s, n let Ms = (1/s)VsV T s, where Vs is an n × s matrix composed of i.i.d. N(0, 1) random variables. Assume n = n(s) and n/s → y ∈ (0, 1) as s → ∞. Then it is shown that the smallest eigenvalue of Ms converges almost surely to $(1 - \sqrt y)^2$ as s → ∞.