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

Let X = (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 = (ρij ) be the p × p sample correlation coefficient matrix of X, and S = (1/n)X(n)(X(n))∗ − X̄X̄∗ be the sample covariance matrix of X, 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 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 converges almost surely to (1 −√c )2.