Point process convergence for the off-diagonal entries of sample covariance matrices. (English) Zbl 1479.60101
Summary: We study point process convergence for sequences of i.i.d. random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a \((2+\delta )\) th moment. We make heavy use of precise large deviation results for sums of i.i.d. random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60G70 | Extreme value theory; extremal stochastic processes |
| 60F10 | Large deviations |
| 60G50 | Sums of independent random variables; random walks |
| 62F05 | Asymptotic properties of parametric tests |
Keywords:
extreme value theory; Gumbel distribution; maximum entry; precise large deviations; sample covariance matrixReferences:
| [1] | Arendarczyk, M. and Dȩbicki, K. (2011). Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli 17 194-210. · Zbl 1284.60074 · doi:10.3150/10-BEJ266 |
| [2] | Auffinger, A., Ben Arous, G. and Péché, S. (2009). Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 45 589-610. · Zbl 1177.15037 · doi:10.1214/08-AIHP188 |
| [3] | Bai, Z. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer Series in Statistics. Springer, New York. · Zbl 1301.60002 · doi:10.1007/978-1-4419-0661-8 |
| [4] | Bickel, P. J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist. 36 2577-2604. · Zbl 1196.62062 · doi:10.1214/08-AOS600 |
| [5] | Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199-227. · Zbl 1132.62040 · doi:10.1214/009053607000000758 |
| [6] | Bun, J., Bouchaud, J.-P. and Potters, M. (2017). Cleaning large correlation matrices: Tools from random matrix theory. Phys. Rep. 666 1-109. · Zbl 1359.15031 · doi:10.1016/j.physrep.2016.10.005 |
| [7] | Cai, T., Liu, W. and Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. J. Amer. Statist. Assoc. 108 265-277. · Zbl 06158341 · doi:10.1080/01621459.2012.758041 |
| [8] | Cai, T. T. and Jiang, T. (2011). Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann. Statist. 39 1496-1525. · Zbl 1220.62066 · doi:10.1214/11-AOS879 |
| [9] | Cai, T. T. and Jiang, T. (2012). Phase transition in limiting distributions of coherence of high-dimensional random matrices. J. Multivariate Anal. 107 24-39. · Zbl 1352.60006 · doi:10.1016/j.jmva.2011.11.008 |
| [10] | Davis, R. A., Heiny, J., Mikosch, T. and Xie, X. (2016). Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series. Extremes 19 517-547. · Zbl 1384.60023 · doi:10.1007/s10687-016-0251-7 |
| [11] | Davis, R. A., Mikosch, T. and Pfaffel, O. (2016). Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series. Stochastic Process. Appl. 126 767-799. · Zbl 1331.60017 · doi:10.1016/j.spa.2015.10.001 |
| [12] | Denisov, D., Dieker, A. B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: The big-jump domain. Ann. Probab. 36 1946-1991. · Zbl 1155.60019 · doi:10.1214/07-AOP382 |
| [13] | Donoho, D. (2000). High-dimensional data analysis: The curses and blessings of dimensionality. Technical Report, Stanford Univ. |
| [14] | Einmahl, U. (1989). Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 20-68. · Zbl 0676.60038 · doi:10.1016/0047-259X(89)90097-3 |
| [15] | El Karoui, N. (2009). Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond. Ann. Appl. Probab. 19 2362-2405. · Zbl 1255.62156 · doi:10.1214/08-AAP548 |
| [16] | Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin. · Zbl 0873.62116 · doi:10.1007/978-3-642-33483-2 |
| [17] | Fyodorov, Y. V. and Bouchaud, J.-P. (2008). Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 372001, 12. · Zbl 1214.82016 · doi:10.1088/1751-8113/41/37/372001 |
| [18] | Heiny, J. (2019). Large correlation matrices: A comparison theorem and its applications. Submitted. |
| [19] | Heiny, J. and Mikosch, T. (2017). Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The iid case. Stochastic Process. Appl. 127 2179-2207. · Zbl 1378.60024 · doi:10.1016/j.spa.2016.10.006 |
| [20] | Jiang, T. (2004). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865-880. · Zbl 1047.60014 · doi:10.1214/105051604000000143 |
| [21] | Jiang, T. and Xie, J. (2019). Limiting behavior of largest entry of random tensor constructed by high-dimensional data. arXiv:1910.12701v1. · Zbl 1471.60031 |
| [22] | Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078 · doi:10.1214/aos/1009210544 |
| [23] | Johnstone, I. M. and Titterington, D. M. (2009). Statistical challenges of high-dimensional data. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 4237-4253. · Zbl 1185.62007 · doi:10.1098/rsta.2009.0159 |
| [24] | Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin; Academic Press [Harcourt Brace Jovanovich, Publishers], London. · Zbl 0544.60053 |
| [25] | Li, D., Liu, W.-D. and Rosalsky, A. (2010). Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Related Fields 148 5-35. · Zbl 1210.62010 · doi:10.1007/s00440-009-0220-z |
| [26] | Li, D., Qi, Y. and Rosalsky, A. (2012). On Jiang’s asymptotic distribution of the largest entry of a sample correlation matrix. J. Multivariate Anal. 111 256-270. · Zbl 1275.62037 · doi:10.1016/j.jmva.2012.04.002 |
| [27] | Li, D. and Rosalsky, A. (2006). Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16 423-447. · Zbl 1098.60034 · doi:10.1214/105051605000000773 |
| [28] | Liu, W.-D., Lin, Z. and Shao, Q.-M. (2008). The asymptotic distribution and Berry-Esseen bound of a new test for independence in high dimension with an application to stochastic optimization. Ann. Appl. Probab. 18 2337-2366. · Zbl 1154.60021 · doi:10.1214/08-AAP527 |
| [29] | Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 (114) 507-536. · Zbl 0152.16101 |
| [30] | Michel, R. (1974). Results on probabilites of moderate deviations. Ann. Probab. 2 349-353. · Zbl 0282.60015 · doi:10.1214/aop/1176996719 |
| [31] | Nagaev, A. V. (1969). Integral limit theorems with regard to large deviations when Cramér’s condition is not satisfied. I. Teor. Veroyatn. Primen. 14 51-63. · Zbl 0172.21901 |
| [32] | Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745-789. · Zbl 0418.60033 |
| [33] | Petrov, V. V. (1972). Sums of Independent Random Variables (in Russian). Izdat. “Nauka”, Moscow. |
| [34] | Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications. · Zbl 0826.60001 |
| [35] | Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York. · Zbl 1152.62029 |
| [36] | Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York. Reprint of the 1987 original. · Zbl 1136.60004 |
| [37] | Rozovskiĭ, L. V. (1993). Probabilities of large deviations on the whole axis. Teor. Veroyatn. Primen. 38 79-109. · Zbl 0801.60021 · doi:10.1137/1138005 |
| [38] | Soshnikov, A. (2004). Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Commun. Probab. 9 82-91. · Zbl 1060.60013 · doi:10.1214/ECP.v9-1112 |
| [39] | Soshnikov, A. (2006). Poisson statistics for the largest eigenvalues in random matrix ensembles. In Mathematical Physics of Quantum Mechanics. Lecture Notes in Physics 690 351-364. Springer, Berlin. · Zbl 1169.15302 · doi:10.1007/3-540-34273-7_26 |
| [40] | Yao, J., Zheng, S. and Bai, Z. (2015). Large Sample Covariance Matrices and High-Dimensional Data Analysis. Cambridge Series in Statistical and Probabilistic Mathematics 39. Cambridge Univ. Press, New York. · Zbl 1380.62011 · doi:10.1017/CBO9781107588080 |
| [41] | Zhou, W. (2007). Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc. 359 5345-5363 · Zbl 1130.60032 · doi:10.1090/S0002-9947-07-04192-X |
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