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Title: Almost sure limit of the smallest eigenvalue of some sample correlation matrices
Authors: Xiao, H.
Zhou, W. 
Keywords: Random matrix
Sample correlation coefficient matrix
Sample covariance matrix
Smallest eigenvalue
Issue Date: Jan-2010
Citation: Xiao, H., Zhou, W. (2010-01). Almost sure limit of the smallest eigenvalue of some sample correlation matrices. Journal of Theoretical Probability 23 (1) : 1-20. ScholarBank@NUS Repository.
Abstract: Let X(n) = (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(n) = (ρij) be the p × p sample correlation coefficient matrix of X(n), and S(n) = (1/n)X(n)(X(n))*-X̄X̄* be the sample covariance matrix of X(n), 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(n) 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(n) converges almost surely to (1-√c)2. © Springer Science+Business Media, LLC 2009.
Source Title: Journal of Theoretical Probability
ISSN: 08949840
DOI: 10.1007/s10959-009-0270-2
Appears in Collections:Staff Publications

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