Skip to main content
Log in

Random Matrix Theory for Heavy-Tailed Time Series

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This paper is a review of recent results for large random matrices with heavy-tailed entries. First, we outline the development of and some classical results in random matrix theory. We focus on large sample covariance matrices, their limiting spectral distributions, and the asymptotic behavior of their largest and smallest eigenvalues and their eigenvectors. The limits significantly depend on the finite or infiniteness of the fourth moment of the entries of the random matrix. We compare the results for these two regimes which give rise to completely different asymptotic theories. Finally, the limits of the extreme eigenvalues of sample correlation matrices are examined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Auffinger, G. Ben Arous, and S. Péché, “Poisson convergence for the largest eigenvalues of heavy tailed random matrices,” Ann. Inst. Henri Poincaré Probab. Stat., 45, No. 3, 589–610 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. Z. Bai, Z. Fang, and Y.-C. Liang, Spectral Theory of Large Dimensional Random Matrices and Its Applications to Wireless Communications and Finance Statistics: Random Matrix Theory and Its Applications, World Scientific (2014).

  3. Z. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, Springer, New York (2010).

  4. Z. Bai and W. Zhou, “Large sample covariance matrices without independence structures in columns,” Stat. Sinica, 18, No. 2, 425–442 (2008).

    MathSciNet  MATH  Google Scholar 

  5. Z. D. Bai, “Methodologies in spectral analysis of large-dimensional random matrices, a review,” Stat. Sinica, 9, No. 3, 611–677 (1999).

    MathSciNet  MATH  Google Scholar 

  6. Z. D. Bai and Y. Q. Yin, “Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix,” Ann. Probab., 21, No. 3, 1275–1294 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Belinschi, A. Dembo, and A. Guionnet, “Spectral measure of heavy tailed band and covariance random matrices,” Commun. Math. Phys., 289, No. 3, 1023–1055 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Ben Arous and A. Guionnet, “The spectrum of heavy tailed random matrices,” Commun. Math. Phys., 278, No. 3, 715–751 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Benaych-Georges and S. Péché, “Localization and delocalization for heavy tailed band matrices,” Ann. Inst. Henri Poincaré Probab. Stat., 50, No. 4, 1385–1403 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Bhatia, Matrix Analysis, Springer-Verlag, New York (1997).

    Book  MATH  Google Scholar 

  11. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge (1987).

    Book  MATH  Google Scholar 

  12. R. A. Davis, J. Heiny, T. Mikosch, and X. Xie, “Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series,” Extremes, 19, No. 3, 517–547 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  13. R. A. Davis, T. Mikosch, and O. Pfaffel, “Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series,” Stoch. Proc. Appl., 126, No. 3, 767–799 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  14. R. A. Davis, O. Pfaffel, and R. Stelzer, “Limit theory for the largest eigenvalues of sample covariance matrices with heavy-tails,” Stoch. Proc. Appl., 124, No. 1, 18–50 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. N. El Karoui, “On the largest eigenvalue of Wishart matrices with identity covariance when n,p and p/n tend to infinity,” Available at http://arxiv.org/abs/math/0309355 (2003).

  16. P. Embrechts and N. Veraverbeke, “Estimates for the probability of ruin with special emphasis on the possibility of large claims,” Insur. Math. Econ., 1, No. 1, 55–72 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, John Wiley & Sons, New York (1966).

    MATH  Google Scholar 

  18. S. Geman, “A limit theorem for the norm of random matrices,” Ann. Probab., 8, No. 2, 252–261 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Giné, F. Götze, and D. M. Mason, “When is the Student t-statistic asymptotically standard normal?” Ann. Probab., 25, No. 3, 1514–1531 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Heiny, Extreme Eigenvalues of Sample Covariance and Correlation Matrices, Ph.D. Thesis, University of Copenhagen (2017).

  21. J. Heiny and T. Mikosch, “Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices,” Stoch. Proc. Appl., 29 (2017).

  22. J. Heiny and T. Mikosch, “The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails,” Submitted for publication (2017).

  23. J. Heiny and T. Mikosch, “Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The i.i.d. case,” Stoch. Proc. Appl., 127, No. 7, 2179–2207 (2017).

  24. A. Janssen, T. Mikosch, R. Mohsen, and X. Xiaolei, “The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model,” Bernoulli, 24, No. 2, 1351–1393 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Jiang, “The limiting distributions of eigenvalues of sample correlation matrices,” Sankhyā, 66, No. 1, 35–48 (2004).

    MathSciNet  MATH  Google Scholar 

  26. I. M. Johnstone, “On the distribution of the largest eigenvalue in principal components analysis,” Ann. Stat., 29, No. 2, 295–327 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  27. S. O’Rourke, V. Vu, and K. Wang, “Eigenvectors of random matrices: a survey,” J. Combin. Theor. Ser. A, 144, 361–442 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Péché, “Universality results for the largest eigenvalues of some sample covariance matrix ensembles,” Probab. Theor. Rel. Fields, 143, No. 3–4, 481–516 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  29. S. I. Resnick, Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York (2007).

    MATH  Google Scholar 

  30. S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer, New York (2008).

    Google Scholar 

  31. M. Rudelson and R. Vershynin, “Delocalization of eigenvectors of random matrices with independent entries,” Duke Math. J., 164, No. 13, 2507–2538 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  32. J. W. Silverstein, “Some limit theorems on the eigenvectors of large-dimensional sample covariance matrices,” J. Multivariate Anal., 15, No. 3, 295–324 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Soshnikov, “Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails,” Electron. Commun. Probab., 9, 82–91 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Soshnikov, “Poisson statistics for the largest eigenvalues in random matrix ensembles,” in: Mathematical Physics of Quantum Mechanics, Springer, Berlin (2006), pp. 351–364.

    MATH  Google Scholar 

  35. K. Tikhomirov “The limit of the smallest singular value of random matrices with i.i.d. entries,” Adv. Math., 284, 1–20 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  36. C. A. Tracy and H. Widom, “Distribution functions for largest eigenvalues and their applications,” in: Proceedings of the International Congress of Mathematicians, Vol. I, Higher Ed. Press, Beijing (2002), pp. 587–596.

  37. E. P. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. Math., 62, No. 2, 548–564 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  38. E. P. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions II,” Ann. Math., 65, No. 2, 203–207 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Wishart, “The generalised product moment distribution in samples from a normal multivariate population,” Biometrika, 32–52 (1928).

  40. H. Xiao and W. Zhou, “Almost sure limit of the smallest eigenvalue of some sample correlation matrices,” J. Theor. Probab., 23, No. 1, 1–20 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Yao, S. Zheng, and Z. Bai, Large Sample Covariance Matrices and High-Dimensional Data Analysis, Cambridge University Press, New York (2015).

    Book  MATH  Google Scholar 

  42. Y. Q. Yin, Z. D. Bai, and P. R. Krishnaiah, “On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix,” Probab. Theor. Relat. Fields, 78, No. 4, 509–521 (1988).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. Heiny.

Additional information

Proceedings of the XXXV International Seminar on Stability Problems for Stochastic Models, Debrecen, Hungary, August 25–29, 2017. Part I

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heiny, J. Random Matrix Theory for Heavy-Tailed Time Series. J Math Sci 237, 652–666 (2019). https://doi.org/10.1007/s10958-019-04191-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-019-04191-3

Navigation