ALMOST SURE LIMIT OF THE SMALLEST EIGENVALUE OF THE SAMPLE CORRELATION MATRIX By Han Xiao and Wang Zhou National University of Singapore

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 ( X )∗ − X̄X̄∗ be the sample covariance matrix of X, where X̄ is the mean vector of the n observations. Assuming that Xij ’s 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 ) as n →∞ and p/n → c ∈ (0 , ∞). We accomplish this by showing that the smallest eigenvalue of S converges almost surely to (1−√c ).