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Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. (English) Zbl 0779.60026
Summary: The authors show that the smallest (if \(p\leq n)\) or the \((p-n+1)\)-th smallest (if \(p>n)\) eigenvalue of a sample covariance matrix of the form \((1/n)XX'\) tends almost surely to the limit \((1-\sqrt y)^ 2\) as \(n\to\infty\) and \(p/n\to y\in(0,\infty)\), where \(X\) is a \(p\times n\) matrix with i.i.d. entries with mean zero, variance 1 and fourth moment finite. Also, as a by-product, it is shown that the almost sure limit of the largest eigenvalue is \((1+\sqrt y)^ 2\), a known result obtained by the authors and P. R. Krishnaiah [Probab. Theory Relat. Fields 78, No. 4, 509–521 (1988; Zbl 0627.62022)]. The present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices.

60F15 Strong limit theorems
62H99 Multivariate analysis
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