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

• Volume: 48, Issue: 2, page 444-476
• ISSN: 0246-0203

top

## Abstract

top
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 + d ⋅ F matrix)−1, where d is a constant and I is an identity matrix.

## How to cite

top

Zheng, Shurong. "Central limit theorems for linear spectral statistics of large dimensional F-matrices." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 444-476. <http://eudml.org/doc/272099>.

@article{Zheng2012,
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 + d ⋅ F matrix)−1, where d is a constant and I is an identity matrix.},
author = {Zheng, Shurong},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {linear spectral statistics; central limit theorem; F-matrix; beta matrix; Fisher matrix; covariance matrices},
language = {eng},
number = {2},
pages = {444-476},
publisher = {Gauthier-Villars},
title = {Central limit theorems for linear spectral statistics of large dimensional F-matrices},
url = {http://eudml.org/doc/272099},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Zheng, Shurong
TI - Central limit theorems for linear spectral statistics of large dimensional F-matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 444
EP - 476
AB - 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 + d ⋅ F matrix)−1, where d is a constant and I is an identity matrix.
LA - eng
KW - linear spectral statistics; central limit theorem; F-matrix; beta matrix; Fisher matrix; covariance matrices
UR - http://eudml.org/doc/272099
ER -

## References

top
1. [1] G. W. Anderson and O. Zeitouni. A CLT for a band matrix model. Probab. Theory Related Fields134 (2006) 283–338. Zbl1084.60014MR2222385
2. [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. Zbl0698.62021
3. [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. Zbl0937.60017MR1617051
4. [4] Z. D. Bai and J. W. Silverstein. Exact separation of eigenvalues of large dimensional sample covariance matrices. Ann. Probab.27 (1999) 1536–1555. Zbl0964.60041MR1733159
5. [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. Zbl1063.60022MR2040792
6. [6] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large-Dimensional Random Matrices, 2nd edition. Springer, New York, 2010. Zbl1301.60002MR2567175
7. [7] Z. D. Bai and J. F. Yao. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli11 (2005) 1059–1092. Zbl1101.60012MR2189081
8. [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. Zbl0657.62058MR847583
9. [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. Zbl1081.82569MR1327898
10. [10] T. Cabanal-Duvillard. Fluctuations de la loi empirique de grande matrices aléatoires. Ann. Inst. H. Poincaré Probab. Statist.37 (2001) 373–402. Zbl1016.15020MR1831988
11. [11] S. Chatterjee. Fluctuations of eigenvalues and second order poincaré inequalities. Probab. Theory Related Fields143 (2009) 1–40. Zbl1152.60024MR2449121
12. [12] O. Costin and J. L. Lebowitz. Gaussian fluctuation in random matrices. Phys. Rev. Lett.75 (1995) 69–72.
13. [13] P. Diaconis and S. N. Evans. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc.353 (2001) 2615–2633. Zbl1008.15013MR1828463
14. [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. Zbl1112.82021MR2239975
15. [15] V. L. Girko. Theory of Random Determinants. Kluwer Academic Publishers, London, 1990. Zbl0704.60003MR1080966
16. [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. Zbl0995.60028MR1899457
17. [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. Theory54 (2008) 3987–4004. Zbl1322.94003MR2450782
18. [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. Zbl1166.15013MR2473651
19. [19] P. Hall and C. C. Heyde. Martingale Limit Theory and Its Application. Academic Press, New York, 1980. Zbl0462.60045MR624435
20. [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. Zbl0987.60039MR1844632
21. [21] S. Israelson. Asymptotic fluctuations of a particle system with singular interaction. Stochastic Process. Appl.93 (2001) 25–56. Zbl1053.60104MR1819483
22. [22] T. Jiang. Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles. Probab. Theory Related Fields144 (2009) 221–246. Zbl1162.60005MR2480790
23. [23] K. Johansson. On random matrices from the classical compact groups. Ann. Math.145 (1997) 519–545. Zbl0883.60010MR1454702
24. [24] K. Johansson. On the fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J.91 (1998) 151–204. Zbl1039.82504MR1487983
25. [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. Zbl1120.62033MR2334195
26. [26] D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal.12 (1982) 1–38. Zbl0491.62021MR650926
27. [27] J. P. Keating and N. C. Snaith. Random matrix theory and ζ(1/2 + it). Comm. Math. Phys.214 (2000) 57–89. Zbl1051.11048MR1794265
28. [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. Zbl0866.15014MR1411619
29. [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. Zbl1100.46040MR2216446
30. [30] K. C. S. Pillai. Percentage points of the largest root of the multivariate beta matrix. Biometrika54 (1967) 189–194. Zbl0149.15803MR215433
31. [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. Zbl0559.62040MR754832
32. [32] B. Ridelury and J. W. Silverstein. Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab.34 (2006) 2118–2143. Zbl1122.15022MR2294978
33. [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. Zbl1130.60030MR2361453
34. [34] J. W. Silverstein. The limiting eigenvalue distribution of a multivariate F-matrix. SIAM J. Math. Anal.16 (1985) 641–646. Zbl0606.62054MR783987
35. [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. Zbl0872.60013MR1345541
36. [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. Zbl0912.15027MR1620151
37. [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. Zbl0930.15025MR1647832
38. [38] A. Soshnikov. Gaussian limits for determinantal random point fields. Ann. Probab.28 (2002) 171–181. Zbl1033.60063MR1894104
39. [39] K. Wieand. Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields123 (2002) 202–224. Zbl1044.15016MR1900322
40. [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. Zbl0531.62018MR727036

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.