Yaskov, P. A. On the spectrum of sample covariance matrices for time series. (English. Russian original) Zbl 1396.62219 Theory Probab. Appl. 62, No. 3, 432-443 (2018); translation from Teor. Veroyatn. Primen. 62, No. 3, 542-555 (2017). Summary: We study the spectrum of the sample covariance matrix corresponding to an \(\mathbb R^p\)-valued time series of length \(n\). Under the assumption \(p/n\rightarrow\rho >0\) conditions are put forward to guarantee the universality property of the limiting spectral distribution of these matrices (it has the same form as in the case of Gaussian time series). These conditions amount to requiring that the quadratic forms of the values of the series be close to its means. Cited in 1 Document MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60B20 Random matrices (probabilistic aspects) 62M15 Inference from stochastic processes and spectral analysis Keywords:random matrices; ample covariance matrices; time series; spectrum PDFBibTeX XMLCite \textit{P. A. Yaskov}, Theory Probab. Appl. 62, No. 3, 432--443 (2018; Zbl 1396.62219); translation from Teor. Veroyatn. Primen. 62, No. 3, 542--555 (2017) Full Text: DOI References: [1] R. Adamczak, Some remarks on the Dozier–Silverstein theorem for random matrices with dependent entries, Random Matrices Theory Appl., 2 (2013), 1250017. · Zbl 1282.60006 [2] S. Anatolyev and P. Yaskov, Asymptotics of diagonal elements of projection matrices under many instruments/regressors, Econometric Theory, 33 (2017), pp. 717–738. · Zbl 1441.62583 [3] Z. D. Bai, Methodologies in spectral analysis of large dimensional random matrices, a review, Statist. Sinica, 9 (1999), pp. 611–677. · Zbl 0949.60077 [4] Z. Bai and W. Zhou, Large sample covariance matrices without independence structures in columns, Statist. Sinica, 18 (2008), pp. 425–442. · Zbl 1135.62009 [5] M. Banna, Limiting spectral distribution of Gram matrices associated with functionals of \(β\)-mixing processes, J. Math. Anal. Appl., 433 (2016), pp. 416–433. · Zbl 1323.60012 [6] M. Banna, F. Merlevède, and M. Peligrad, On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries, Stochastic Process. Appl., 125 (2015), pp. 2700–2726. · Zbl 1328.60083 [7] M. Bhattacharjee and A. Bose, Large sample behaviour of high dimensional autocovariance matrices, Ann. Statist., 44 (2016), pp. 598–628. · Zbl 1343.62053 [8] N. El Karoui, Concentration of measure and spectra of random matrices: Applications to correlation matrices\(,\) elliptical distributions and beyond, Ann. Appl. Probab., 19 (2009), pp. 2362–2405. · Zbl 1255.62156 [9] N. El Karoui, High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation, Ann. Statist., 38 (2010), pp. 3487–3566. · Zbl 1274.62365 [10] V. L. Girko and A. K. Gupta, Asymptotic behavior of spectral function of empirical covariance matrices, Random Oper. Stochastic Equations, 2 (1994), pp. 43–60. · Zbl 0827.62050 [11] H. Liu, A. Aue, and D. Paul, On the Mar\vcenko–Pastur law for linear time series, Ann. Statist., 43 (2015), pp. 675–712. · Zbl 1312.62080 [12] J. R. Magnus, The moments of products of quadratic forms in normal variables, Statist. Neerlandica, 32 (1978), pp. 201–210. · Zbl 0406.62031 [13] V. A. Marchenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR-Sb., 1 (1967), pp. 457–483. · Zbl 0162.22501 [14] O. Pfaffel and E. Schlemm, Limiting spectral distribution of a new random matrix model with dependence across rows and columns, Linear Algebra Appl., 436 (2012), pp. 2966–2979. · Zbl 1244.15026 [15] T. Tao, Topics in Random Matrix Theory, Grad. Stud. Math. 132, Amer. Math. Soc., Providence, RI, 2012. · Zbl 1256.15020 [16] P. Yaskov, Variance inequalities for quadratic forms with applications, Math. Methods Statist., 24 (2015), pp. 309–319. · Zbl 1334.60096 [17] P. Yaskov, Necessary and sufficient conditions for the Marchenko–Pastur theorem, Electron. Commun. Probab., 21 (2016), 73. · Zbl 1352.60008 [18] P. Yaskov, A short proof of the Marchenko–Pastur theorem, C. R. Math. Acad. Sci. Paris, 354 (2016), pp. 319–322. · Zbl 1380.60017 [19] P. Yaskov, LLN for quadratic forms of long memory time series and its applications in random matrix theory, J. Theoret. Probab., to appear. · Zbl 1428.60051 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.