Moment Bounds for Large Autocovariance Matrices Under Dependence

  • Fang HanEmail author
  • Yicheng Li


The goal of this paper is to obtain expectation bounds for the deviation of large sample autocovariance matrices from their means under weak data dependence. While the accuracy of covariance matrix estimation corresponding to independent data has been well understood, much less is known in the case of dependent data. We make a step toward filling this gap and establish deviation bounds that depend only on the parameters controlling the “intrinsic dimension” of the data up to some logarithmic terms. Our results have immediate impacts on high-dimensional time-series analysis, and we apply them to high-dimensional linear VAR(d) model, vector-valued ARCH model, and a model used in Banna et al. (Random Matrices Theory Appl 5(2):1650006, 2016).


Autocovariance matrix Effective rank Weak dependence \(\tau \)-mixing 

Mathematics Subject Classification (2010)

60E15 60F10 



  1. 1.
    Andrews, D.W.: Non-strong mixing autoregressive processes. J. Appl. Probab. 21(4), 930–934 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bai, Z., Yin, Y.: Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21(3), 1275–1294 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banna, M., Merlevède, F., Youssef, P.: Bernstein-type inequality for a class of dependent random matrices. Random Matrices Theory Appl. 5(2), 1650006 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berbee, H.C.: Random Walks with Stationary Increments and Renewal Theory, vol. 112. Mathematisch Centrum, Amsterdam (1979)zbMATHGoogle Scholar
  5. 5.
    Blinn, J.: Consider the lowly \(2 \times 2\) matrix. IEEE Comput. Graph. Appl. 16(2), 82–88 (1996)CrossRefGoogle Scholar
  6. 6.
    Brand, M.: Fast low-rank modifications of the thin singular value decomposition. Linear Algebra Appl. 415(1), 20–30 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brillinger, D.R.: Time Series: Data Analysis and Theory. Siam, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bunea, F., Xiao, L.: On the sample covariance matrix estimator of reduced effective rank population matrices, with applications to fPCA. Bernoulli 21(2), 1200–1230 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, J., Guo, B., Yao, Q.: Principal component analysis for second-order stationary vector time series. Ann. Stat. 46(5), 2094–2124 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, X., Xu, M., Wu, W.B.: Covariance and precision matrix estimation for high-dimensional time series. Ann. Stat. 41(6), 2994–3021 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Davis, C., Kahan, W.M.: The rotation of eigenvectors by a perturbation. iii. SIAM J. Numer. Anal. 7(1), 1–46 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dedecker, J., Doukhan, P., Lang, G., Leon, J., Louhichi, S., Prieur, C.: Weak Dependence: With Examples and Applications. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dedecker, J., Prieur, C.: Coupling for \(\tau \)-dependent sequences and applications. J. Theor. Probab. 17(4), 861–885 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Han, F., Liu, H.: ECA: high-dimensional elliptical component analysis in non-gaussian distributionsigh-dimensional elliptical component analysis in non-Gaussian distributions. J. Am. Stat. Assoc. 113(521), 252–268 (2018)CrossRefzbMATHGoogle Scholar
  15. 15.
    Koltchinskii, V., Lounici, K.: Concentration inequalities and moment bounds for sample covariance operators. Bernoulli 23(1), 110–133 (2017a)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Koltchinskii, V., Lounici, K.: New asymptotic results in principal component analysis. Sankhya A 79(2), 254–297 (2017b)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Koltchinskii, V., Lounici, K.: Normal approximation and concentration of spectral projectors of sample covariance. Ann. Stat. 45(1), 121–157 (2017c)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, W., Xiao, H., Wu, W.B.: Probability and moment inequalities under dependence. Stat. Sin. 23(3), 1257–1272 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lounici, K.: High-dimensional covariance matrix estimation with missing observations. Bernoulli 20(3), 1029–1058 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mendelson, S.: Empirical processes with a bounded \(\psi _1\) diameter. Geom. Funct. Anal. 20(4), 988–1027 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mendelson, S., Paouris, G.: On the singular values of random matrices. J. Eur. Math. Soc. 16, 823–834 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Merlevède, F., Peligrad, M., Rio, E.: Bernstein inequality and moderate deviations under strong mixing conditions. High Dimensional Probability V: The Luminy Volume, pp. 273–292. Institute of Mathematical Statistics, Beachwood (2009)CrossRefGoogle Scholar
  23. 23.
    Merlevède, F., Peligrad, M., Rio, E.: A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probab. Theory Relat. Fields 151(3), 435–474 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Oliveira, R.: Sums of random Hermitian matrices and an inequality by Rudelson. Electron. Commun. Probab. 15, 203–212 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Petz, D.: A survey of certain trace inequalities. Banach Center Publ. 30(1), 287–298 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rudelson, M.: Random vectors in the isotropic position. J. Funct. Anal. 164(1), 60–72 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Slepian, D.: The one-sided barrier problem for Gaussian noise. Bell Syst. Techn. J. 41(2), 463–501 (1962)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Srivastava, N., Vershynin, R.: Covariance estimation for distributions with \(2+\epsilon \) moments. Ann. Probab. 41(5), 3081–3111 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Talagrand, M.: Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  30. 30.
    Tikhomirov, K.: Sample covariance matrices of heavy-tailed distributions. Int. Math. Res. Not. 2018(20), 6254–6289 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tropp, J.A.: An introduction to matrix concentration inequalities. Found. Trends Mach. Learn. 8(1–2), 1–230 (2015)CrossRefzbMATHGoogle Scholar
  32. 32.
    van Handel, R.: Structured random matrices. Convexity and Concentration, vol. 161, pp. 107–156. Springer, Berlin (2017)CrossRefGoogle Scholar
  33. 33.
    Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. Compressed Sensing, pp. 210–268. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  34. 34.
    Wu, W.B.: Nonlinear system theory: another look at dependence. Proc. Natl. Acad. Sci. U. S. A. 102(40), 14150–14154 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wu, W.B., Wu, Y.N.: Performance bounds for parameter estimates of high-dimensional linear models with correlated errors. Electron. J. Stat. 10(1), 352–379 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WashingtonSeattleUSA

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