## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Central limit theorems for linear spectral statistics of large dimensional F-matrices

Shurong Zheng

#### Abstract

In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the F matrix. In this paper, we establish the central limit theorems with explicit expressions of means and covariance functions for the linear spectral statistics of the large dimensional F matrix, where the dimension p of the two samples tends to infinity proportionally to the sample sizes (n1, n2). Moreover, the assumptions of the i.i.d. structures of arrays (Xjk)p×n1, (Yjk)p×n2 and the restriction of the fourth moments equaling 2 or 3 made in Bai and Silverstein (Ann. Probab. 32 (2004) 553–605) are relaxed to that arrays (Xjk)p×n1 and (Yjk)p×n2 are independent respectively but not necessarily identically distributed except for a common fourth moment for each array. As a consequence, we obtain the central limit theorems for the linear spectral statistics of the beta matrix that is of the form (I + dF matrix)−1, where d is a constant and I is an identity matrix.

#### Résumé

Dans beaucoup d’applications, on cherche à effectuer une inférence statistique sur des paramètres définis à partir de la mesure spectrale d’une F-matrice, matrice obtenue comme le produit d’une matrice de covariance du tableau de variables indépendantes (Xjk)p×n1 et de l’inverse d’une autre matrice de covariance (Yjk)p×n2. Les variables sont soient toutes réelles soient complexes. Il est donc utile d’étudier les distributions asymptotiques des estimateurs de ces paramètres associés à la F-matrice. Dans cet article, nous établissons des théorèmes centraux limites pour les statistiques linéaires du spectre de la F-matrice dans la limite où p, n1, n2 tendent vers l’infini en restant de même ordre, et donnons des formules exactes pour leurs moyennes et covariances. De plus, l’hypothèse que les variables (Xjk)p×n1 et (Yjk)p×n2 sont i.i.d. et la restriction que le quatrième moment est égal à 2 ou 3 comme dans Bai et Silverstein (Ann. Probab. 32 (2004) 553–605) sont affaiblies de la manière suivante; les coefficients (Xjk)p×n1 et (Yjk)p×n2 sont indépendants mais non nécessairement équidistribués, pourvu qu’ils aient le même quatrième moment dans chaque tableau. Par conséquent, nous obtenons le théorème de la limite centrale pour les statistiques linéaires de la matrice beta qui est de la forme (I + dF matrix)−1, où d est une constante et I la matrice identité.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 444-476.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148207

Digital Object Identifier
doi:10.1214/11-AIHP414

Mathematical Reviews number (MathSciNet)
MR2954263

Zentralblatt MATH identifier
1251.15039

#### Citation

Zheng, Shurong. Central limit theorems for linear spectral statistics of large dimensional F -matrices. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 444--476. doi:10.1214/11-AIHP414. https://projecteuclid.org/euclid.aihp/1334148207

#### References

• [1] G. W. Anderson and O. Zeitouni. A CLT for a band matrix model. Probab. Theory Related Fields 134 (2006) 283–338.
• [2] Z. D. Bai. A note on asymptotic joint distribution of the eigenvalues of a noncentral multivariate F-matrix. Technical report, Central for Multivariate Analysis, Univ. Pittsburgh, 1984.
• [3] Z. D. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of large dimensional random matrices. Ann. Probab. 26 (1998) 316–345.
• [4] Z. D. Bai and J. W. Silverstein. Exact separation of eigenvalues of large dimensional sample covariance matrices. Ann. Probab. 27 (1999) 1536–1555.
• [5] Z. D. Bai and J. W. Silverstein. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 (2004) 553–605.
• [6] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large-Dimensional Random Matrices, 2nd edition. Springer, New York, 2010.
• [7] Z. D. Bai and J. F. Yao. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 (2005) 1059–1092.
• [8] Z. D. Bai, Y. Q. Yin and P. R. Krishnaiah. On LSD of product of two random matrices when the underlying distribution is isotropic. J. Multivariate Anal. 19 (1986) 189–200.
• [9] A. Boutet De Monvel, L. Pastur and M. Shcherbina. On the statistical mechanics approach in the random matrix theory, integrated density of states. J. Stat. Phys. 79 (1995) 585–611.
• [10] T. Cabanal-Duvillard. Fluctuations de la loi empirique de grande matrices aléatoires. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 373–402.
• [11] S. Chatterjee. Fluctuations of eigenvalues and second order poincaré inequalities. Probab. Theory Related Fields 143 (2009) 1–40.
• [12] O. Costin and J. L. Lebowitz. Gaussian fluctuation in random matrices. Phys. Rev. Lett. 75 (1995) 69–72.
• [13] P. Diaconis and S. N. Evans. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 (2001) 2615–2633.
• [14] I. Dumitriu and A. Edelman. Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models. J. Math. Phys. 47 (2006) 063302.
• [15] V. L. Girko. Theory of Random Determinants. Kluwer Academic Publishers, London, 1990.
• [16] A. Guionnet. Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 341–384.
• [17] W. Hachem, O. Khorunzhiy, P. Loubaton, J. Najim and L. Pastur. A new approach for capacity analysis of large dimensional multi-antenna channels. IEEE Trans. Inform. Theory 54 (2008) 3987–4004.
• [18] W. Hachem, P. Loubaton and J. Najim. A CLT for information-theoretical statistics of Gram random matrices with a given variance profile. Ann. Appl. Probab. 18 (2008) 2071–2130.
• [19] P. Hall and C. C. Heyde. Martingale Limit Theory and Its Application. Academic Press, New York, 1980.
• [20] C. P. Hughes, J. P. Keating and N. O’Connell. On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220 (2001) 429–451.
• [21] S. Israelson. Asymptotic fluctuations of a particle system with singular interaction. Stochastic Process. Appl. 93 (2001) 25–56.
• [22] T. Jiang. Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles. Probab. Theory Related Fields 144 (2009) 221–246.
• [23] K. Johansson. On random matrices from the classical compact groups. Ann. Math. 145 (1997) 519–545.
• [24] K. Johansson. On the fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1998) 151–204.
• [25] I. M. Johnstone. High dimensional statistical inference and random matrices. In International Congress of Mathematicians, Vol. I 307–333. Eur. Math. Soc. Zürich, Switzerland, 2007.
• [26] D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 (1982) 1–38.
• [27] J. P. Keating and N. C. Snaith. Random matrix theory and ζ(1/2 + it). Comm. Math. Phys. 214 (2000) 57–89.
• [28] A. M. Khorunzhy, B. A. Knoruzhenko and L. A. Pastur. Asymptotic properties of large random matrices with independent entrices. J. Math. Phys. 37 (1996) 5033–5060.
• [29] J. A. Mingo and R. Speicher. Second order freeness and fluctuations of random matrices I, Gaussian and Wishart matrices and cyclic Fock spaces. J. Funct. Anal. 235 (2006) 226–270.
• [30] K. C. S. Pillai. Percentage points of the largest root of the multivariate beta matrix. Biometrika 54 (1967) 189–194.
• [31] K. C. S. Pillai and B. N. Flury. Percentage points of the largest characteristic root of the multivariate beta matrix. Comm. Statist. 13 (1984) 2199–2237.
• [32] B. Ridelury and J. W. Silverstein. Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. 34 (2006) 2118–2143.
• [33] B. Rider and B. Virág. The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. 2007 (2007) Article ID rnm006.
• [34] J. W. Silverstein. The limiting eigenvalue distribution of a multivariate F-matrix. SIAM J. Math. Anal. 16 (1985) 641–646.
• [35] J. W. Silverstein and S. I. Choi. Analysis of the limiting spectral distribution of large dimensional random matrices. J. Multivariate Anal. 54 (1995) 295–309.
• [36] Y. A. Sinaǐ and A. Soshnikov. Central limit theorems for traces of large random matrices with independent entries. Bol. Soc. Brasil. Mat. 29 (1998) 1–24.
• [37] Y. A. Sinaǐ and A. Soshnikov. A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl. 32 (1998) 114–131.
• [38] A. Soshnikov. Gaussian limits for determinantal random point fields. Ann. Probab. 28 (2002) 171–181.
• [39] K. Wieand. Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields 123 (2002) 202–224.
• [40] Y. Q. Yin, Z. D. Bai and P. R. Krishnaiah. Limiting behavior of the eigenvalues of a multivariate F-matrix. J. Multivariate Anal. 13 (1983) 508–516.