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

摘要

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 be the sample covariance matrix of X (n), where 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 as n→∞ and p→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S (n) converges almost surely to . Random matrix - Sample correlation coefficient matrix - Sample covariance matrix - Smallest eigenvalueMathematics Subject Classification (2000) 60H15 - 62H99 Wang Zhou was partially supported by an NUS grant R-155-000-083-112.