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Generalized four moment theorem and an application to CLT for spiked eigenvalues of high-dimensional covariance matrices. (English) Zbl 1475.60014
Summary: We consider a more generalized spiked covariance matrix, which is a general non-negative definite matrix with the spiked eigenvalues scattered into spaces of a few bulks and the largest ones allowed to tend to infinity. The study is split into two cases by whether the maximum absolute value of the eigenvector of the corresponding spikes tends to zero or not. On one hand, if it is zero, a Generalized Four Moment Theorem (G4MT) is proposed by relaxing the matching of the 3rd and the 4th moment to the tail probability decaying with certain rate, which shows the universality of the asymptotic law for the spiked eigenvalues of the generalized spiked covariance model. On the other hand, if it is not zero, the matches of the third and fourth moments in usual four moment theorem are weakened to only requiring the match of the 4th moment. Moreover, by applying the results to the Central Limit Theorem (CLT) for the spiked eigenvalues of the generalized spiked covariance model, we successively remove the restrictive condition of block wise diagonal assumption on the population covariance matrix in the previous works. This condition implies an unrealistic fact that the spiked eigenvalues and bulked eigenvalues are generated by independent variables, respectively. Thus, the new CLT will have much better application domain.

MSC:
 60B20 Random matrices (probabilistic aspects) 60F05 Central limit and other weak theorems
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 [1] Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica 70 191-221. Zentralblatt MATH: 1103.91399 Digital Object Identifier: doi:10.1111/1468-0262.00273 · Zbl 1103.91399 [2] Bai, Z., Choi, K.P. and Fujikoshi, Y. (2018). Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis. Ann. Statist. 46 1050-1076. Zentralblatt MATH: 1395.62119 Digital Object Identifier: doi:10.1214/17-AOS1577 Project Euclid: euclid.aos/1525313075 · Zbl 1395.62119 [3] Bai, Z. and Ding, X. (2012). Estimation of spiked eigenvalues in spiked models. Random Matrices Theory Appl. 1 1150011, 21. Zentralblatt MATH: 1251.15037 Digital Object Identifier: doi:10.1142/S2010326311500110 · Zbl 1251.15037 [4] Bai, Z. and Yao, J. (2008). Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. Henri Poincaré Probab. Stat. 44 447-474. Zentralblatt MATH: 1274.62129 Digital Object Identifier: doi:10.1214/07-AIHP118 Project Euclid: euclid.aihp/1211819420 · Zbl 1274.62129 [5] Bai, Z. and Yao, J. (2012). On sample eigenvalues in a generalized spiked population model. J. Multivariate Anal. 106 167-177. Zentralblatt MATH: 1301.62049 Digital Object Identifier: doi:10.1016/j.jmva.2011.10.009 · Zbl 1301.62049 [6] Bai, Z.D., Miao, B.Q., Rao and Radbakrisbna, C. (1991). Estimation of directions of arrival of signals: Asymptotic results. In Advances in Spectrum Analysis and Array Processing, Vol. I 327-347. [7] Bai, Z.D. and Silverstein, J.W. (1999). Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27 1536-1555. Zentralblatt MATH: 0964.60041 Digital Object Identifier: doi:10.1214/aop/1022677458 Project Euclid: euclid.aop/1022677458 · Zbl 0964.60041 [8] Bai, Z.D. and Silverstein, J.W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 553-605. Zentralblatt MATH: 1063.60022 Digital Object Identifier: doi:10.1214/aop/1078415845 Project Euclid: euclid.aop/1078415845 · Zbl 1063.60022 [9] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643-1697. Zentralblatt MATH: 1086.15022 Digital Object Identifier: doi:10.1214/009117905000000233 Project Euclid: euclid.aop/1127395869 · Zbl 1086.15022 [10] Baik, J. and Silverstein, J.W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382-1408. Zentralblatt MATH: 1220.15011 Digital Object Identifier: doi:10.1016/j.jmva.2005.08.003 · Zbl 1220.15011 [11] Ben Arous, G. and Péché, S. (2005). Universality of local eigenvalue statistics for some sample covariance matrices. Comm. Pure Appl. Math. 58 1316-1357. Zentralblatt MATH: 1075.62014 Digital Object Identifier: doi:10.1002/cpa.20070 · Zbl 1075.62014 [12] Cai, T.T., Han, X. and Pan, G.M. (2019). Limiting laws for divergent spiked eigenvalues and largest non-spiked eigenvalue of sample covariance matrices. Ann. Statist. To appear. http://arxiv.org/abs/1711.00217v2. Mathematical Reviews (MathSciNet): MR4124322 Zentralblatt MATH: 07241590 Digital Object Identifier: doi:10.1214/18-AOS1798 · Zbl 1456.62113 [13] Dyson, F.J. (1970). Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19 235-250. Zentralblatt MATH: 0221.62019 Digital Object Identifier: doi:10.1007/BF01646824 Project Euclid: euclid.cmp/1103842703 · Zbl 0221.62019 [14] Erdos, L., Péché, S., Ramírez, J.A., Schlein, B. and Yau, H.-T. (2010). Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 895-925. · Zbl 1216.15025 [15] Erdos, L., Ramírez, J.A., Schlein, B. and Yau, H.-T. (2010). Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 526-603. · Zbl 1225.15034 [16] Hoyle, D.C. and Rattray, M. (2004). Principal-component-analysis eigenvalue spectra from data with symmetry-breaking structure. Phys. Rev. E 69 026124. Zentralblatt MATH: 1078.68121 · Zbl 1078.68121 [17] Hu, J. and Bai, Z.D. (2014). Estimation of directions of arrival of signals: Asymptotic results. Sci. China Math. 57. [18] Jiang, D. and Bai, Z. (2020). Supplement to “Generalized four moment theorem and an application to CLT for spiked eigenvalues of high-dimensional covariance matrices.” https://doi.org/10.3150/20-BEJ1237SUPP [19] Johansson, K. (2001). Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 683-705. Zentralblatt MATH: 0978.15020 Digital Object Identifier: doi:10.1007/s002200000328 · Zbl 0978.15020 [20] Johnstone, I.M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. Zentralblatt MATH: 1016.62078 Digital Object Identifier: doi:10.1214/aos/1009210544 Project Euclid: euclid.aos/1009210544 · Zbl 1016.62078 [21] Mehta, M.L. (1967). Random Matrices and the Statistical Theory of Energy Levels. New York: Academic Press. Zentralblatt MATH: 0925.60011 · Zbl 0925.60011 [22] Onatski, A. (2009). Testing hypotheses about the numbers of factors in large factor models. Econometrica 77 1447-1479. Zentralblatt MATH: 1182.62180 Digital Object Identifier: doi:10.3982/ECTA6964 · Zbl 1182.62180 [23] Passemier, D. and Yao, J.-F. (2012). On determining the number of spikes in a high-dimensional spiked population model. Random Matrices Theory Appl. 1 1150002, 19. Zentralblatt MATH: 1244.62028 Digital Object Identifier: doi:10.1142/S201032631150002X · Zbl 1244.62028 [24] Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 1617-1642. Zentralblatt MATH: 1134.62029 · Zbl 1134.62029 [25] Skorohod, A.V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1 261-290. [26] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697-733. Zentralblatt MATH: 1062.82502 Digital Object Identifier: doi:10.1007/s002200050743 · Zbl 1062.82502 [27] Tao, T. and Vu, V. (2015). Random matrices: Universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43 782-874. Zentralblatt MATH: 1316.15042 Digital Object Identifier: doi:10.1214/13-AOP876 Project Euclid: euclid.aop/1422885575 · Zbl 1316.15042 [28] Wang, W. and Fan, J. (2017). Asymptotics of empirical eigenstructure for high dimensional spiked covariance. Ann. Statist. 45 1342-1374. Zentralblatt MATH: 1373.62299 Digital Object Identifier: doi:10.1214/16-AOS1487 Project Euclid: euclid.aos/1497319697 · Zbl 1373.62299 [29] Wigner, E. · Zbl 0085.13203
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