We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavy-tailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.
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Anderson, T.W.: Asymptotic theory for principal component analysis. Ann. Math. Stat. 34, 122–148 (1963)
Auffinger, A., Ben Arous, G., Péché, S.: Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 45(3), 589–610 (2009)
Bai, Z., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices, second ed. Springer Series in Statistics. Springer, New York (2010)
Bai, Z.D., Silverstein, J.W., Yin, Y.Q.: A note on the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivar. Anal. 26(2), 166–168 (1988)
Belinschi, S., Dembo, A., Guionnet, A.: Spectral measure of heavy tailed band and covariance random matrices. Comm. Math. Phys. 289(3), 1023–1055 (2009)
Belitskii, G.R., Lyubich, Y.I.: Matrix Norms and their Applications vol. 36 of Operator Theory Advances and Applications. Birkhäuser Verlag, Basel (1988)
Ben Arous, G., Guionnet, A.: The spectrum of heavy tailed random matrices. Comm. Math. Phys. 278(3), 715–751 (2008)
Bhatia, R.: Matrix Analysis, Vol. 169 of Graduate Texts in Mathematics. Springer-Verlag, New York (1997)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation Vol. 27 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1987)
Cline, D.B.H., And Hsing, T.: Large Deviation Probabilities for Sums of Random Variables with Heavy Or Subexponential Tails. Technical Report Statistics Dept., Texas A&M University (1998)
Davis, R.A., Mikosch, T., Pfaffel, O.: Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series. Stoch. Process. Appl. (2015)
Davis, R.A., Pfaffel, O., Stelzer, R.: Limit theory for the largest eigenvalues of sample covariance matrices with heavy-tails. Stoch. Process. Appl. 124(1), 18–50 (2014)
de Haan, L., Ferreira, A.: Extreme Value Theory an Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)
Denisov, D., Dieker, A.B., Shneer, V.: Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36(5), 1946–1991 (2008)
El Karoui, N.: On the largest eigenvalue of wishart matrices with identity covariance when n,p and p/n tend to infinity. arXiv:math/0309355 (2003)
Embrechts, P., Goldie, C.M.: On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29(2), 243–256 (1980)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance, Vol. 33 of Applications of Mathematics (New York). Springer, Berlin (1997)
Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1(1), 55–72 (1982)
Feller, W.: An Introduction to Probability Theory and Its Applications Vol. II. John Wiley & Sons, Inc., New York-London-Sydney (1966)
Geman, S.: A limit theorem for the norm of random matrices. Ann. Probab. 8(2), 252–261 (1980)
Heiny, J., Mikosch, T.: Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: the iid case. To appear (2015)
Heiny, J., Mikosch, T., Davis, R. A.: Limit theory for the singular values of the sample autocovariance matrix function of multivariate time series. Work in progress (2015)
Horn, R.A., Johnson, C.R. Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215(3), 683–705 (2001)
Johnstone, I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29(2), 295–327 (2001)
Lam, C., Yao, Q.: Factor modeling for high-dimensional time series: inference for the number of factors. Ann. Stat. 40(2), 694–726 (2012)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. John Wiley & Sons Inc., New York (1982). Wiley Series in Probability and Mathematical Statistics
Nagaev, S. V.: Large deviations of sums of independent random variables. Ann. Probab. 7(5), 745–789 (1979)
Resnick, S.I.: Heavy-Tail Phenomena Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007)
Resnick, S. I.: Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York (2008). Reprint of the 1987 original
Shores, T.S.: Applied Linear Algebra and Matrix Analysis Undergraduate Texts in Mathematics. Springer, New York (2007)
Soshnikov, A.: Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9, 82–91 (2004). electronic
Soshnikov, A.: Poisson statistics for the largest eigenvalues in random matrix ensembles. In: Mathematical physics of quantum mechanics vol. 690 of Lecture Notes in Phys, pp 351–364. Springer, Berlin (2006)
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298(2), 549–572 (2010)
Tracy, C.A., Widom, H.: Distribution functions for largest eigenvalues and their applications. In: Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pp 587–596. Higher Ed. Press, Beijing (2002)
Richard Davis was supported by ARO MURI grant W911NF-12-1-0385. Thomas Mikosch’s and Johannes Heiny’s research is partly supported by the Danish Research Council Grant DFF-4002-00435 “Large random matrices with heavy tails and dependence”.
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Davis, R.A., Heiny, J., Mikosch, T. et al. Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series. Extremes 19, 517–547 (2016). https://doi.org/10.1007/s10687-016-0251-7
- Regular variation
- Sample covariance matrix
- Dependent entries
- Largest eigenvalues
- Point process convergence
- Cluster poisson limit
- Infinite variance stable limit
- Fréchet distribution
AMS 2000 Subject Classifications
- Primary 60B20; Secondary 60F05