Almost sure limit of the smallest eigenvalue of some sample correlation matrices

Abstract

Let X(n) = (Xij) be a p × n data matrix, where the n columns form a random sample of size n from a certain p-dimensional distribution. Let R(n) = (ρij) be the p × p sample correlation coefficient matrix of X(n), and S(n) = (1/n)X(n)(X(n))*-X̄X̄* be the sample covariance matrix of X(n), where X̄ is the mean vector of the n observations. Assuming that Xij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R(n) converges almost surely to the limit (1-√c)2 as n → ∞ and p/n → c ∈ (0,∞). We accomplish this by showing that the smallest eigenvalue of S(n) converges almost surely to (1-√c)2. © Springer Science+Business Media, LLC 2009.