Some limit theorems for the eigenvalues of a sample covariance matrix

@article{Jonsson1982SomeLT,
  title={Some limit theorems for the eigenvalues of a sample covariance matrix},
  author={Dag Jonsson},
  journal={Journal of Multivariate Analysis},
  year={1982},
  volume={12},
  pages={1-38}
}
  • Dag Jonsson
  • Published 1 March 1982
  • Mathematics
  • Journal of Multivariate Analysis
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References

SHOWING 1-10 OF 15 REFERENCES
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…
Asymptotic distribution of eigenvalues of random matrices
The impetus for this paper comes mainly from work done in recent years by a number of physicists on a statistical theory of spectra. The book by M. L. Mehta [10] and the collection of reprints edited…
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…
Statistical theory of energy levels and random matrices in physics
In this paper the physical aspects of the statistical theory of the energy levels of complex physical systems and their relation to the mathematical theory of random matrices are discussed. After a…
Linear Statistical Inference and its Applications.
Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation.…
Spectral Analysis of Networks with Random Topologies
A class of neural models is introduced in which the topology of the neural network has been generated by a controlled probability model. It is shown that the resulting linear operator has a spectral…
On Wigner's semicircle law for the eigenvalues of random matrices
Some Recent Developments on Real Multivariate Distributions* *Part of this work was sponsored by the Air Force Flight Dynamics Laboratory, Air Force Systems Command, United States Air Force under Grant AFOSR 77-3239.
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.
An Introduction to Multivariate Statistical Analysis
Preface to the Third Edition.Preface to the Second Edition.Preface to the First Edition.1. Introduction.2. The Multivariate Normal Distribution.3. Estimation of the Mean Vector and the Covariance…
...
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