Limiting spectral distribution of XX ' matrices
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, p. 677-707
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.
Une des méthodes pour obtenir la limite des distributions spectrales (LSD) des grandes matrices aléatoires est la fameuse méthode des moments, basée sur la formule des traces. Son succès a été clairement établi pour différents types de matrices telles que les matrices de Wigner et les matrices de covariance. Dans un article récent, Bryc, Dembo et Jiang [Ann. Probab. 34 (2006) 1-38] ont obtenu la LSD pour des matrices de Toeplitz et de Hankel en utilisant cette méthode. Ils arrivent à estimer les traces des moments de telles matrices en séparant les différents termes par classes d'équivalence et en reliant les asymptotiques des dénombrements afférents avec les calculs de certains volumes. Bose et Sen [Electron. J. Probab. 13 (2008) 588-628] ont développé cette idée et ont donné un cadre général pour traiter de matrices symmétriques dont les entrées viennent d'une suite indépendante. Dans cet article, nous généralisons cette approche pour considérer des matrices de la form où X est une matrice p×n avec des entrées réelles. Nous démontrons un résultat général d'existence de la LSD de telles matrices, correctement recentrées et rééchelonnées, quand p et n tendent vers l'infini de telle façon que p/n tende vers y∈(0, ∞). Par exemple, nous montrons l'existence de la LSD quand X est la matrice asymétrique de Hankel, de Toeplitz, circulante ou circulante inverse. En particulier, quand y=0, les limites correspondent à celles obtenues par Bryc, Dembo et Jiang [Ann. Probab. 34 (2006) 1-38]. Sinon, nous obtenons de nouvelles lois limites pour lesquelles aucune expression explicite n'est connue. Nous étudions ces lois par quelques simulations.
DOI : https://doi.org/10.1214/09-AIHP329
Classification:  60F05,  60F15,  62E20,  60G57
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
@article{AIHPB_2010__46_3_677_0,
     author = {Bose, Arup and Gangopadhyay, Sreela and Sen, Arnab},
     title = {Limiting spectral distribution of $XX^{\prime }$ matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {3},
     year = {2010},
     pages = {677-707},
     doi = {10.1214/09-AIHP329},
     zbl = {1226.60007},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_3_677_0}
}
Bose, Arup; Gangopadhyay, Sreela; Sen, Arnab. Limiting spectral distribution of $XX^{\prime }$ matrices. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 3, pp. 677-707. doi : 10.1214/09-AIHP329. http://www.numdam.org/item/AIHPB_2010__46_3_677_0/

[1] Z. D. Bai. Methodologies in spectral analysis of large dimensional random matrices, a review. Statist. Sinica 9 (1999) 611-677 (with discussions). | MR 1711663 | Zbl 0949.60077

[2] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices. Science Press, Beijing, 2006. | Zbl 1196.60002

[3] Z. D. Bai and Y. Q. Yin. Convergence to the semicircle law. Ann. Probab. 16 (1988) 863-875. | MR 929083 | Zbl 0648.60030

[4] R. Bhatia. Matrix Analysis. Springer, New York, 1997. | MR 1477662 | Zbl 0863.15001

[5] A. Bose and J. Mitra. Limiting spectral distribution of a special circulant. Statist. Probab. Lett. 60 (2002) 111-120. | MR 1945684 | Zbl 1014.60038

[6] A. Bose and A. Sen. Another look at the moment method for large dimensional random matrices. Electron. J. Probab. 13 (2008) 588-628. | MR 2399292 | Zbl 1190.60013

[7] W. Bryc, A. Dembo and T. Jiang. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006) 1-38. | MR 2206341 | Zbl 1094.15009

[8] W. Feller. An Introduction to Probability Theory and Its Applications 2. Wiley, New York, 1966. | MR 210154 | Zbl 0219.60003

[9] U. Grenander and J. W. Silverstein. Spectral analysis of networks with random topologies. SIAM J. Appl. Math. 32 (1977) 499-519. | MR 476178 | Zbl 0355.94043

[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. | MR 2167641 | Zbl 1086.15024

[11] D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 (1982) 1-38. | MR 650926 | Zbl 0491.62021

[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. | Zbl 0978.15500

[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. | MR 208649 | Zbl 0152.16101

[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. | MR 2337145 | Zbl 1126.15030

[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] A. M. Tulino and S. Verdu. Random Matrix Theory and Wireless Communications. Now Publishers Inc., 2004. | Zbl 1133.94014

[17] K. W. Wachter. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 (1978) 1-18. | MR 467894 | Zbl 0374.60039

[18] E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955) 548-564. | MR 77805 | Zbl 0067.08403

[19] E. P. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958) 325-327. | MR 95527 | Zbl 0085.13203

[20] J. Wishart. The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A (1928) 32-52. | JFM 54.0565.02

[21] Y. Q. Yin. Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20 (1986) 50-68. | MR 862241 | Zbl 0614.62060

[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. | MR 816299 | Zbl 0584.62029