Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices

Let X (n)=(X ij ) 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)=(ρ ij ) be the p×p sample correlation coefficient matrix of X (n), and $S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}$ be the sample covariance matrix of X (n), where $\bar{X}$ is the mean vector of the n observations. Assuming that X ij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R (n) converges almost surely to the limit $(1-\sqrt{c}\,)^{2}$ as n→∞ and p/n→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S (n) converges almost surely to $(1-\sqrt{c}\,)^{2}$ .