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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.

MSC:
 60F15 Strong limit theorems 62H99 Multivariate analysis
Citations:
Zbl 0641.62015; Zbl 0627.62022
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