Limiting spectral distribution of XX' matrices

Arup Bose; Sreela Gangopadhyay; Arnab Sen

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 3, page 677-707
  • ISSN: 0246-0203

Abstract

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The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well-known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalence classes and relating the limits of the counts to certain volume calculations. Bose and Sen [Electron. J. Probab.13 (2008) 588–628] have developed this method further and have provided a general framework which deals with symmetric matrices with entries coming from an independent sequence. In this article we enlarge the scope of the above approach to consider matrices of the form where X is a p×n matrix with real entries. We establish some general results on the existence of the spectral distribution of such matrices, appropriately centered and scaled, when p→∞ and n=n(p)→∞ and p/n→y with 0≤y<∞. As examples we show the existence of the spectral distribution when X is taken to be the appropriate asymmetric Hankel, Toeplitz, circulant and reverse circulant matrices. In particular, when y=0, the limits for all these matrices coincide and is the same as the limit for the symmetric Toeplitz derived in Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38]. In other cases, we obtain new limiting spectral distributions for which no closed form expressions are known. We demonstrate the nature of these limits through some simulation results.

How to cite

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Bose, Arup, Gangopadhyay, Sreela, and Sen, Arnab. "Limiting spectral distribution of XX' matrices." Annales de l'I.H.P. Probabilités et statistiques 46.3 (2010): 677-707. <http://eudml.org/doc/242616>.

@article{Bose2010,
abstract = {The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well-known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalence classes and relating the limits of the counts to certain volume calculations. Bose and Sen [Electron. J. Probab.13 (2008) 588–628] have developed this method further and have provided a general framework which deals with symmetric matrices with entries coming from an independent sequence. In this article we enlarge the scope of the above approach to consider matrices of the form where X is a p×n matrix with real entries. We establish some general results on the existence of the spectral distribution of such matrices, appropriately centered and scaled, when p→∞ and n=n(p)→∞ and p/n→y with 0≤y&lt;∞. As examples we show the existence of the spectral distribution when X is taken to be the appropriate asymmetric Hankel, Toeplitz, circulant and reverse circulant matrices. In particular, when y=0, the limits for all these matrices coincide and is the same as the limit for the symmetric Toeplitz derived in Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38]. In other cases, we obtain new limiting spectral distributions for which no closed form expressions are known. We demonstrate the nature of these limits through some simulation results.},
author = {Bose, Arup, Gangopadhyay, Sreela, Sen, Arnab},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {large dimensional random matrix; eigenvalues; sample covariance matrix; Toeplitz matrix; Hankel matrix; circulant matrix; reverse circulant matrix; spectral distribution; bounded Lipschitz metric; limiting spectral distribution; moment method; volume method; almost sure convergence; convergence in distribution},
language = {eng},
number = {3},
pages = {677-707},
publisher = {Gauthier-Villars},
title = {Limiting spectral distribution of XX' matrices},
url = {http://eudml.org/doc/242616},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Bose, Arup
AU - Gangopadhyay, Sreela
AU - Sen, Arnab
TI - Limiting spectral distribution of XX' matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 677
EP - 707
AB - The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well-known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the trace by splitting the relevant sets into equivalence classes and relating the limits of the counts to certain volume calculations. Bose and Sen [Electron. J. Probab.13 (2008) 588–628] have developed this method further and have provided a general framework which deals with symmetric matrices with entries coming from an independent sequence. In this article we enlarge the scope of the above approach to consider matrices of the form where X is a p×n matrix with real entries. We establish some general results on the existence of the spectral distribution of such matrices, appropriately centered and scaled, when p→∞ and n=n(p)→∞ and p/n→y with 0≤y&lt;∞. As examples we show the existence of the spectral distribution when X is taken to be the appropriate asymmetric Hankel, Toeplitz, circulant and reverse circulant matrices. In particular, when y=0, the limits for all these matrices coincide and is the same as the limit for the symmetric Toeplitz derived in Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38]. In other cases, we obtain new limiting spectral distributions for which no closed form expressions are known. We demonstrate the nature of these limits through some simulation results.
LA - eng
KW - large dimensional random matrix; eigenvalues; sample covariance matrix; Toeplitz matrix; Hankel matrix; circulant matrix; reverse circulant matrix; spectral distribution; bounded Lipschitz metric; limiting spectral distribution; moment method; volume method; almost sure convergence; convergence in distribution
UR - http://eudml.org/doc/242616
ER -

References

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