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Gaussian Fluctuations in Complex Sample Covariance Matrices


 
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1. Title Title of document Gaussian Fluctuations in Complex Sample Covariance Matrices
 
2. Creator Author's name, affiliation, country Zhonggen Su; Zhejiang University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Central limit theorem; the Costin-Lebowitz-Soshnikov theorem; Eigenvalues; RH problems; Sample covariance matrices
 
3. Subject Subject classification AMS subject classification (2000): 15A52; 60F05.
 
4. Description Abstract Let $X=(X_{i,j})_{m\times n}, m\ge n$, be a complex Gaussian random matrix with mean zero and variance $\frac 1n$, let $S=X^*X$ be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when $\frac mn\rightarrow \gamma\ge 1$ as $n\rightarrow\infty$. Under certain conditions on $k$, we prove the central limit theorem holds true for the $k$-th largest eigenvalues $\lambda_{(k)}$ as $k$ tends to infinity as $n\rightarrow\infty$. The proof is largely based on the Costin-Lebowitz-Soshnikov argument and the asymptotic estimates for the expectation and variance of the number of eigenvalues in an interval. The standard technique for the RH problem is used to compute the exact formula and asymptotic properties for the mean density of eigenvalues. As a by-product, we obtain a convergence speed of the mean density of eigenvalues to the Marchenko-Pastur distribution density under the condition $|\frac mn-\gamma|=O(\frac 1n)$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Supported partly by NSF of China (No.10371109) and the Royal Society K.C.Wong Educational Foundation.
 
7. Date (YYYY-MM-DD) 2006-12-17
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/378
 
10. Identifier Digital Object Identifier 10.1214/EJP.v11-378
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 11
 
12. Language English=en
 
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