DESCRIBING THE BEHAVIOR OF EIGENVECTORS OF RANDOM MATRICES USING SEQUENCES OF MEASURES ON ORTHOGONAL GROUPS

@article{Silverstein1981DESCRIBINGTB,
  title={DESCRIBING THE BEHAVIOR OF EIGENVECTORS OF RANDOM MATRICES USING SEQUENCES OF MEASURES ON ORTHOGONAL GROUPS},
  author={Jack W. Silverstein},
  journal={Siam Journal on Mathematical Analysis},
  year={1981},
  volume={12},
  pages={274-281}
}
  • J. W. Silverstein
  • Published 1 March 1981
  • Mathematics
  • Siam Journal on Mathematical Analysis
A conjecture has previously been made on the chaotic behavior of the eigenvectors of a class of n-dimensional random matrices, where n is very large [J. Silverstein, SIAM J. Appl. Math., 37 (1979), pp. 235–245]. Evidence supporting the conjecture has been given in the form of two limit theorems, as $n \to \infty $, relating the random matrices to matrices formed from the Haar measure, $h_n $, on the orthogonal group $\mathcal{O}_n $.The present paper considers a reformulation of the conjecture… 
Functional CLT of eigenvectors for large sample covariance matrices
In order to investigate property of the eigenvector matrix of sample covariance matrix $$\mathbf {S}_n$$Sn, in this paper, we establish the central limit theorem of linear spectral statistics
Random truncations of Haar distributed matrices and bridges
TLDR
It is proved that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process, on the way to other interesting convergences.
Bridges and random truncations of random matrices
TLDR
The deterministic truncation of U is replaced by a random one, in which each row is chosen with probability s (respectively, column) independently, to prove that the corresponding two-parameter process, after centering and normalization by n-1/2 converges to a Gaussian process.
EIGENVECTORS OF SAMPLE COVARIANCE MATRICES : UNIVERSALITY OF GLOBAL FLUCTUATIONS
  • Ali
  • Mathematics
  • 2013
In this paper, we prove a universality result of convergence for a bivariate random process defined by the eigenvectors of a sample covariance matrix. Let Vn = (vij)i≤n, j≤m be a n ×m random matrix,
Eigenvectors of Sample Covariance Matrices: Universality of global fluctuations
In this paper, we prove a universality result of convergence for a bivariate random process defined by the eigenvectors of a sample covariance matrix. Let Vn = (vij)i n,j m be a n◊ m random matrix,
Asymptotic properties of eigenmatrices of a large sample covariance matrix
TLDR
This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of $S_n$ is asymptotically close to that of a Haar-distributed unitary matrix.
Study on Asymptotic Properties of Eigenvectors of Large Sample Covariance Matrix
Let Sn = 1 n XnX ∗ n where Xn = {Xij} is p × n matrix with i.i.d. complex standardized entries having finite fourth moment. Different from previous literature in which eigenvalues are of interest to
Truncations of Haar distributed matrices, traces and bivariate Brownian bridges
Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor}
Multivariate Statistical Analysis and Random Matrix Theory
TLDR
This chapter is concerned with a number of issues in statistics that have a grouptheoretic flavor, including numerical sampling techniques, multivariate statistical analysis, and theory/numerical procedures associated with random orthogonal, positive definite, unitary, and Hermitian matrices.
...
...

References

SHOWING 1-10 OF 14 REFERENCES
ON THE RANDOMNESS OF EIGENVECTORS GENERATED FROM NETWORKS WITH RANDOM TOPOLOGIES
A model for the generation of neural connections at birth led to the study of W, a random, symmetric, nonnegative definite linear operator defined on a finite, but very large, dimensional Euclidean
The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements
This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The
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
Characteristic Vectors of Bordered Matrices with Infinite Dimensions I
The statistical properties of the characteristic values of a matrix the elements of which show a normal (Gaussian) distribution are well known (cf. [6] Chapter XI) and have been derived, rather
On the Distribution of the Roots of Certain Symmetric Matrices
TLDR
The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.
Weak convergence of probability measures and random functions in the function space D [0,∞)
This paper extends the theory of weak convergence of probability measures and random functions in the function space D [0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞)
Convergence of Probability Measures
Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
...
...