CLT FOR LINEAR SPECTRAL STATISTICS OF LARGE-DIMENSIONAL SAMPLE COVARIANCE MATRICES

@inproceedings{Bai2008CLTFL,
  title={CLT FOR LINEAR SPECTRAL STATISTICS OF LARGE-DIMENSIONAL SAMPLE COVARIANCE MATRICES},
  author={Zhidong Bai and Jack W. Silverstein},
  year={2008}
}
Let Bn = (1/N)T 1/2 n XnX∗ nT 1/2 n where Xn = (Xij ) is n × N with i.i.d. complex standardized entries having finite fourth moment, and T 1/2 n is a Hermitian square root of the nonnegative definite Hermitian matrix Tn. The limiting behavior, as n → ∞ with n/N approaching a positive constant, of functionals of the eigenvalues of Bn, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of Bn, it is known that these linear spectral… 
On asymptotics of eigenvectors of large sample covariance matrix
Let {X ij }, i, j = ..., be a double array of i.i.d. complex random variables with EX 11 = 0, E|X 11 | 2 = 1 and E|X 11 | 4 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the
J un 2 01 9 Joint CLT for top eigenvalues of sample covariance matrices of separable high dimensional long memory processes
For N, n ∈ N, consider the sample covariance matrix SN (T ) = 1 N XX ∗ from a data set X = C 1/2 N ZT 1/2 n , where Z = (Zi,j) is a N × n matrix having i.i.d. entries with mean zero and variance one,
On Asymptotic Expansion and CLT of Linear Eigenvalue Statistics for Sample Covariance Matrices When N/M → 0
We study the renormalized real sample covariance matrixH = XX/ √ MN− √ M/N with N/M → 0 as N,M → ∞ in this paper. And we always assume M = M(N). Here X = [Xjk]M×N is an M × N real random matrix with
CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES
We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent
Comparison between two types of large sample covariance matrices
Let {X ij }, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX 11 = µ, E|X 11 − µ| 2 = 1 and E|X 11 | 4 < ∞. Consider sample
ON ASYMPTOTIC EXPANSION AND CENTRAL LIMIT THEOREM OF LINEAR EIGENVALUE STATISTICS FOR SAMPLE COVARIANCE MATRICES WHEN N / M → 0
Abstract. We study the renormalized real sample covariance matrixH = XTX/ √ MN−M/N withN/M → 0 asN,M → ∞. We always assumeM = M(N). HereX = [Xjk]M×N is anM×N real random matrix with independent
Logarithmic law of large random correlation matrix
Consider a random vector y = Σ1/2x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ1/2 is a deterministic p × p matrix such
Central limit theorems for linear spectral statistics of large dimensional F-matrices
In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the
Large deviations of spread measures for Gaussian matrices
TLDR
The statistics of atypically small log-determinants is shown to be driven by the split-off of the smallest eigenvalue, leading to an abrupt change in the large deviation speed.
...
...

References

SHOWING 1-10 OF 22 REFERENCES
EXACT SEPARATION OF EIGENVALUES OF LARGE DIMENSIONAL SAMPLE COVARIANCE MATRICES
Let B n = (1/N)T 1/2 n X n X* n T n 1/2 where X n is n × N with i.i.d. complex standardized entries having finite fourth moment, and T n 1/2 is a Hermitian square root of the nonnegative definite
Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices
Let X be n - N containing i.i.d. complex entries with E X11 - EX112 = 1, and T an n - n random Hermitian nonnegative definite, independent of X. Assume, almost surely, as n --> [infinity], the
Linear functionals of eigenvalues of random matrices
Let Mn be a random n×n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces
Limiting spectral distribution for a class of random matrices
Analysis of the limiting spectral distribution of large dimensional random matrices
Results on the analytic behavior of the limiting spectral distribution of matrices of sample covariance type, studied in Marcenko and Pastur [2] and Yin [8], are derived. Through an equation defining
Signal detection via spectral theory of large dimensional random matrices
TLDR
The theoretical analysis presented focuses on the splitting of the spectrum of sample covariance matrix into noise and signal eigenvalues and it is shown that when the number of sensors is large thenumber of signals can be estimated with a sample size considerably less than that required by previous approaches.
DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES
In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of
Central limit theorem for traces of large random symmetric matrices with independent matrix elements
AbstractWe study Wigner ensembles of symmetric random matricesA=(aij),i, j=1,...,n with matrix elementsaij,i≤j being independent symmetrically distributed random variables $$a_{ij} = a_{ji} =
A HIGH DIMENSIONAL TWO SAMPLE SIGNIFICANCE TEST
0. Summary. The classical multivariate 2 sample significance test based on Hotelling's T2 is undefined when the number k of variables exceeds the number of within sample degrees of freedom available
...
...