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

Han Xiao () and Wang Zhou ()
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Han Xiao: The University of Chicago
Wang Zhou: National University of Singapore

Journal of Theoretical Probability, 2010, vol. 23, issue 1, 1-20

Abstract: 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 $(1-\sqrt{c}\,)^{2}$ .

Keywords: Random matrix; Sample correlation coefficient matrix; Sample covariance matrix; Smallest eigenvalue; 60H15; 62H99 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10959-009-0270-2

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