Abstract
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 \((1-\sqrt{c}\,)^{2}\) .
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Wang Zhou was partially supported by an NUS grant R-155-000-083-112.
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Xiao, H., Zhou, W. Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices. J Theor Probab 23, 1–20 (2010). https://doi.org/10.1007/s10959-009-0270-2
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DOI: https://doi.org/10.1007/s10959-009-0270-2