Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

Sean O’Rourke () and Alexander Soshnikov ()
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Sean O’Rourke: Yale University
Alexander Soshnikov: University of California, Davis

Journal of Theoretical Probability, 2015, vol. 28, issue 2, 726-744

Abstract: Abstract Let $$M_n$$ M n be an $$n \times n$$ n × n Wigner or sample covariance random matrix, and let $$\mu _1(M_n), \mu _2(M_n), \ldots , \mu _n(M_n)$$ μ 1 ( M n ) , μ 2 ( M n ) , … , μ n ( M n ) denote the randomly ordered eigenvalues of $$M_n$$ M n . We study the fluctuations of the partial linear eigenvalue statistics $$\begin{aligned} \sum _{i=1}^{n-k} f(\mu _i(M_n)) \end{aligned}$$ ∑ i = 1 n − k f ( μ i ( M n ) ) as $$n \rightarrow \infty $$ n → ∞ for sufficiently nice test functions $$f$$ f . We consider both the cases when $$k$$ k is fixed and when $$\min \{k,n-k\}$$ min { k , n − k } tends to infinity with $$n$$ n .

Keywords: Random matrix theory; Wigner matrices; Sample covariance matrices; Limit laws; Central limit theorems; 60B20; 60F05 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-013-0491-2

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