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Gaussian fluctuations for non-Hermitian random matrix ensembles. (English) Zbl 1122.15022

An ensemble of \(N \times N\) non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and mean-square one is considered. Z. D. Bai [Ann. Probab. 25, 494–529 (1997; Zbl 0871.62018)] has proved so far the most general result that the circular law holds for ensembles of independent entries such that the common entry distribution possesses a bounded density and finite \(({\text 4}+ \epsilon)\) moment.
In this paper the fluctuations from the circular law are studied in the class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. Attention is paid to linear spectral statistics of the form \(X_N(f) = \sum_{k=1}^N f(\lambda_k)\), where \(\lambda_1, \lambda_2, \dots, \lambda_N\) are ensemble eigenvalues. \(X_N(f)\) is given by \(X_N(f) = N \int f(\lambda) \mu_N (d \lambda)\), where \(\mu_N (d \lambda)\) is the empirical spectral distribution and the test test function \(f\) is analytic in a neighborhood of the disk \(|z| \leq {\text 4}\). The central limit theorem for such an ensemble is established.
The proof is based on the work of Z. D. Bai and J. W. Silverstein [Ann. Probab. 32, 553–605 (2004; Zbl 1063.60022)] and the used method is fairly robust in terms of the underlying distribution of the matrix entries but does not require the analyticity of the test function \(f\) in \(X_N(f)\).

MSC:

15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
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