Abstract
We obtain a weak law of large numbers for quadratic forms of a stationary regular time series, imposing no rate of convergence to zero of its covariance function. We show how this law can be applied in proving universality properties of limiting spectral distributions of sample covariance matrices. In particular, we give another derivation of a recent result of Merlevède and Peligrad, who studied sample covariance matrices generated by IID samples of long memory time series.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adamczak, R.: Some remarks on the Dozier–Silverstein theorem for random matrices with dependent entries. Random Matrices Theory Appl. 2, 46 (2013)
Anatolyev, S., Yaskov, P.: Asymptotics of diagonal elements of projection matrices under many instruments/regressors. Econom. Theory 33(3), 717–738 (2017)
Bhansalia, R.J., Giraitis, L., Kokoszka, P.S.: Approximations and limit theory for quadratic forms of linear processes. Stoch. Proc. Appl. 117(1), 71–95 (2007)
Bai, Z., Zhou, W.: Large sample covariance matrices without independence structures in columns. Stat. Sin. 18, 425–442 (2008)
Beran, J., Feng, Y., Gosh, S., Kulik, R.: Long-Memory Processes: Probabilistic Properties and Statistical Methods. Springer, New York (2013)
Borovskikh, Y.V., Korolyuk, V.S.: Martingale Approximations. VSP BV, Utrecht (1997)
De Jong, R.M.: Law large numbers for dependent heterogeneous processes. Econom. Theory 11(2), 347–358 (1995)
Hall, P.: On the \(L_p\) convergence of sums of independent random variables. Math. Proc. Camb. Philos. Soc. 82, 439–446 (1977)
Girko, V., Gupta, A.K.: Asymptotic behavior of spectral function of empirical covariance matrices. Random Oper. Stoch. Eqs. 2(1), 44–60 (1994)
Götze, F., Tikhomirov, A.N.: Asymptotic distribution of quadratic forms. Ann. Probab. 27, 1072–1098 (1999)
Marchenko, V.A., Pastur, L.A.: Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72, 507–536 (1967)
Marshall, M.W., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York (2011)
Merlevède, F., Peligrad, M.: On the empirical spectral distribution for matrices with long memory and independent rows. Stoch. Proc. Appl. 126(9), 2734–2760 (2016)
Pajor, A., Pastur, L.: On the limiting empirical measure of the sum of rank one matrices with log-concave distribution. Studio Math. 195, 11–29 (2009)
Pastur, L., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, vol. 171. American Mathematical Society, Providence (2011)
Shao, X., Wu, W.B.: A limit theorem for quadratic forms and its applications. Econom. Theory 23, 930–951 (2007)
Shiryaev, A.N.: Probability. Springer, New York (1996)
Silverstein, J.: Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. J. Multivariate Anal. 55, 331–339 (1995)
Tao, T.: Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)
Wu, W.B., Xiao, H.: Covariance matrix estimation for stationary time series. Ann. Stat. 40, 466–493 (2012)
Yao, J.: A note on a Marchenko–Pastur type theorem for time series. Stat. Probab. Lett. 82, 22–28 (2012)
Yaskov, P.: A short proof of the Marchenko–Pastur theorem. Comptes Rendus Mathematique 354, 319–322 (2016)
Yaskov, P.: Necessary and sufficient conditions for the Marchenko–Pastur theorem. Electron. Commun. Probab. (2016). doi:10.1214/16-ECP4748
Yaskov, P.: Variance inequalities for quadratic forms with applications. Math. Methods Stat. 24(4), 309–319 (2015)
Acknowledgements
The author thanks an anonymous reviewer for constructive comments that improved the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the Russian Science Foundation under Grant 14-21-00162.
Rights and permissions
About this article
Cite this article
Yaskov, P. LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory. J Theor Probab 31, 2032–2055 (2018). https://doi.org/10.1007/s10959-017-0767-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-017-0767-z