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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  1. [1] Z. D. Bai. Methodologies in spectral analysis of large dimensional random matrices, a review. Statist. Sinica 9 (1999) 611–677 (with discussions). Zbl0949.60077MR1711663
  2. [2] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices. Science Press, Beijing, 2006. Zbl1196.60002
  3. [3] Z. D. Bai and Y. Q. Yin. Convergence to the semicircle law. Ann. Probab. 16 (1988) 863–875. Zbl0648.60030MR929083
  4. [4] R. Bhatia. Matrix Analysis. Springer, New York, 1997. Zbl0863.15001MR1477662
  5. [5] A. Bose and J. Mitra. Limiting spectral distribution of a special circulant. Statist. Probab. Lett. 60 (2002) 111–120. Zbl1014.60038MR1945684
  6. [6] A. Bose and A. Sen. Another look at the moment method for large dimensional random matrices. Electron. J. Probab. 13 (2008) 588–628. Zbl1190.60013MR2399292
  7. [7] W. Bryc, A. Dembo and T. Jiang. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006) 1–38. Zbl1094.15009MR2206341
  8. [8] W. Feller. An Introduction to Probability Theory and Its Applications 2. Wiley, New York, 1966. Zbl0219.60003MR210154
  9. [9] U. Grenander and J. W. Silverstein. Spectral analysis of networks with random topologies. SIAM J. Appl. Math. 32 (1977) 499–519. Zbl0355.94043MR476178
  10. [10] C. Hammond and S. J. Miller. Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theoret. Probab. 18 (2005) 537–566. Zbl1086.15024MR2167641
  11. [11] D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 (1982) 1–38. Zbl0491.62021MR650926
  12. [12] E. Kaltofen. Asymptotically fast solution of Toeplitz-like singular linear systems. In ISSAC 297–304. ACM, New York, 1994. Available at http://www4.ncsu.edu/~kaltofen/bibliography/94/Ka94_issac.pdf. Zbl0978.15500
  13. [13] V. A. Marčenko and L. A. Pastur. Distribution of eigenvalues for some sets of random matrices. Mat. Sb. (N.S.) 72 (1967) 507–536. Zbl0152.16101MR208649
  14. [14] A. Massey, S. J. Miller and J. Sinsheimer. Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices. J. Theoret. Probab. 20 (2007) 637–662. Zbl1126.15030MR2337145
  15. [15] J. W. Silverstein and A. M. Tulino. Theory of large dimensional random matrices for engineers. IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications 458–464, 2006. 
  16. [16] A. M. Tulino and S. Verdu. Random Matrix Theory and Wireless Communications. Now Publishers Inc., 2004. Zbl1133.94014
  17. [17] K. W. Wachter. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 (1978) 1–18. Zbl0374.60039MR467894
  18. [18] E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955) 548–564. Zbl0067.08403MR77805
  19. [19] E. P. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958) 325–327. Zbl0085.13203MR95527
  20. [20] J. Wishart. The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A (1928) 32–52. Zbl54.0565.02JFM54.0565.02
  21. [21] Y. Q. Yin. Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20 (1986) 50–68. Zbl0614.62060MR862241
  22. [22] Y. Q. Yin and P. R. Krishnaiah. Limit theorem for the eigenvalues of the sample covariance matrix when the underlying distribution is isotropic. Theory Probab. Appl. 30 (1985) 810–816. Zbl0584.62029MR816299

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