TRACY-WIDOM DISTRIBUTION FOR THE LARGEST EIGENVALUE OF REAL SAMPLE COVARIANCE MATRICES WITH GENERAL POPULATION

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We consider sample covariance matrices of the form Q = (Sigma X-1/2)((EX)-X-1/2)*, where the sample X is an M x N random matrix whose entries are real independent random variables with variance 1/N and where Sigma is an M x M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest resealed eigenvalue of Q when both M and N tend to infinity with N/M -> d is an element of (0, infinity). For a large class of populations Sigma in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that Sigma is diagonal and that the entries of X have a sub-exponential decay.
Publisher
INST MATHEMATICAL STATISTICS
Issue Date
2016-12
Language
English
Article Type
Article
Keywords

WIGNER MATRICES; UNIVERSALITY; STATISTICS; LIMIT; ENSEMBLES; FLUCTUATIONS; SIGNALS; EDGE

Citation

ANNALS OF APPLIED PROBABILITY, v.26, no.6, pp.3786 - 3839

ISSN
1050-5164
DOI
10.1214/16-AAP1193
URI
http://hdl.handle.net/10203/218734
Appears in Collection
MA-Journal Papers(저널논문)
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