Eigenvectors of some large sample covariance matrix ensembles
We consider sample covariance matrices where X N is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and Σ N is a N × N positive definite matrix. In addition we assume that the spectral measure of Σ N almost surely converges to some limiting probability distribution as N...
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Published in | Probability theory and related fields Vol. 151; no. 1-2; pp. 233 - 264 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer-Verlag
01.10.2011
Springer Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | We consider sample covariance matrices
where
X
N
is a
N
×
p
real or complex matrix with i.i.d. entries with finite 12th moment and Σ
N
is a
N
×
N
positive definite matrix. In addition we assume that the spectral measure of Σ
N
almost surely converges to some limiting probability distribution as
N
→ ∞ and
p
/
N
→ γ > 0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type
where
I
is the identity matrix,
g
is a bounded function and
z
is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse. |
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ISSN: | 0178-8051 1432-2064 |
DOI: | 10.1007/s00440-010-0298-3 |