## Rammebevilling DFF-4002-00435

The Danish Research Council (DFF - Natur og Univers) has granted 1.9 million kroner to the research project

## Large random matrices with heavy tails and dependence

for the period from September 1, 2014, until August 31, 2017.

• Principal investigator
• Thomas Mikosch (University of Copenhagen, Department of Mathematics)
• Collaborators
• Richard A. Davis (Columbia University, Department of Statistics) Olivier Wintenberger (University Pierre & Marie Curie Paris, Department of Mathematics) Olivier Wintenberger is visiting the Department of Mathematics at the University of Copenhagen in the period from September 1, 2014, until August 31, 2016.

• PhD students at the University of Copenhagen affiliated with the project:
• Johannes Heiny (since March 1, 2014) The PhD project of Johannes Heiny is supported in part (2/3) by this research project and in part (1/3) by the Department of Mathematics of the University of Copenhagen. See here for his CV.

In the classical multivariate statistics or time series setting, data consist of n observations of p-dimensional random vectors, where p is fixed and relatively small compared to the sample size n. With the recent advent of large data sets, the dimension p can be large relative to the sample size and hence standard asymptotics, assuming p is fixed relative to n may provide misleading results.

Structure in multivariate data is often summarized by the sample covariance matrix. For example, principal component analysis, extracts principal component vectors corresponding to the largest eigenvalues. Consequently, there is a need to study asymptotics of the largest eigenvalues of the sample covariance matrix. In the case of p fixed and the (p x n) data matrix consists of iid N(0,1) observations, Anderson (1963) showed that the largest eigenvalue is asymptotically normal. In a now seminal paper, Johnstone (2001) showed that if p=p(n) grows with the sample size n at a suitable rate, then the largest eigenvalue, suitable normalized, converges to the Tracy-Widom distribution. Johnstone's result has been generalized by Tao and Vu (2012), where only 4 moments are needed to determine the limit.

The theory for the largest eigenvalues of sample covariance matrices based on heavy-tailed and dependent data is not as well developed as in the light-tailed iid case. Davis, Mikosch and Pfaffel (2013) study the asymptotic behavior of the largest eigenvalues of the sample covariance matrices of a multivariate time series. The time series is assumed to be heavy-tailed and linearly dependent in time and across the components. Allowing dependence between the rows can appreciably impact the limit behavior of the largest eigenvalues. Instead of obtaining a Poisson process as the limit of the extreme eigenvalues, one now gets a "cluster" Poisson process . That is, the limit can be described by a Poisson point process in which each point produces a ``cluster'' of points. Interestingly, the limit point process is identical to the limit point process derived by Davis and Resnick (1985) for the extremes of a linear process.

Davis et al. (2013) make use of the theory of multivariate regular variation which is most flexible for describing heavy tails of random structures. They consider asymptotic properties of sample covariance matrices of time series with regularly varying components with infinite fourth moment and derive the limit structure of the largest eigenvalues when both the dimension and the sample size tend to infinity simultaneously. Furthermore they prove results about the joint convergence of the largest eigenvalues and the trace of the sample covariance matrix, about the spectral gap, the spacings and other continuous functionals acting on the point process of the scaled eigenvalues.

The research project aims at the following objectives:

• Study of the eigenspaces corresponding to the limiting eigenvalues of sample covariance matrices of multivariate non-linear heavy-tailed (regularly varying) time series.
• Development of large deviations techniques for the largest eigenvalues in the sample covariance matrix for heavy-tailed time series.
• Extension of the results from the sample covariance to sample autocovariance matrices.
• Application of the results to large real-life data sets, in particular from finance.
• Poster by J. Heiny at the European Actuarial Journal Conference in Vienna, September 2014.
• Pdf-file of the poster
• Seminar at Bocconi University Milan, given by TM in September 2014.
• Plenary talk by TM at CLAPEM Conference in Cartagena in September 2014.
• Talk by TM at Workshop on Risk, Climate and the Environment, given by TM at Columbia University New York, December 2014.
• Talk by TM at Conference in Honor of Herold Dehling, given by TM at Bochum University, January 2015.
• Talk by TM at Conference in Honor of Yu.V. Prohorov, given by TM at Steklov Institute of the Academy of Science and Lomonosov State University Moscow, February 2015.
• Talk by TM at Conference in Honor of Nick Bingham, given by TM at Imperial College London, March 2015.
• Talk by TM at Conference in Honor of Paul Doukhan, given by TM at Institut H. Poincar\'e Paris, May 2015.
• Talk by TM at the Workshop on Recent Developments in Statistics for Complex Dependent Data, given by TM in Lokkum (Germany), August 2015.
• Plenary Lecture by TM at European Young Statisticians Meeting, given by TM at Charles University Prague, August 2015.
• Talk by TM at Yu.V. Linnik Centennial Conference, given by TM at Euler International Mathematical Institute and Steklov Institute of the Russian Academy of Sciences, St. Petersburg, September 2015.
• Talk by TM at International Conference on Applied Probablity and Computational Methods in Applied Sciences, given by TM at Shanghai Center for Mathematical Sciences, November 2015.
• Talk by TM at Sid Resnick Gala on the Occasion of the 70th birthday of Sid I. Resnick, given by TM on Wall Street Campus of Cornell University, New York, December 2015.
• Talk by TM at Nomura Seminar at Mathematics Department, University of Oxford, March 2016.
• Talk by TM at Workshop on Dependence, Stability and Extremes. Fields Institute, Toronto, May 2016.
• Lecture Series by TM on Heavy-Tail Phenomena at University Marie & Pierre Curie, Paris, June 2016.
• Pleanary talk by TM at 3rd Conference of the International Society for Non-Parametric Statistics, Avignon, June 2016.
• Talk by TM at Conference on the Occasion of Soeren Asmussen's 70th Birthday, Ilulissat (Greenland), August 2016.
• Talk by TM at Conference on the Occasion of Alexander Borovkov's 85th Birthday, Novosibirsk, August 2016.
• Talk by TM at Workshop on Stochastic Networks, Bedlowo (Poland), September 2016.
• Talk by TM at Conference on the Occasion of Valeri Buldygin's 70th Birthday, Kiev Polytechnical Institute, October 2016.
• Talk by TM at Statistics Seminar of the University of Edinburgh, October 2016.
• Talk by TM at Statistics Seminar of the University of Nicosia, November 2016.
• PhD Course by TM on Time Series and Extremes, Bogota, November 2016.
• Defense of PhD thesis "Extreme Eigenvalues of Sample Covariance and Correlation Matrices" by Johannes Heiny on February 14, 2017.
• Hashorva, E., Mikosch, T. and Embrechts, P. (2014) Aggregation of log-linear risks. J. Appl. Probab., 51A, 203-212. See here.
• Mikosch, T. and Zhao, Y. (2015) The integrated periodogram of a dependent extremal event sequence. Stoch. Proc. Appl., 125, 3126-3169. See here.
• Dieker, T. and Mikosch, T.(2015) Exact simulation of a Brown-Resnick random field. Extremes, 18, 301-314. See here.
• Matsui, M. and Mikosch, T.(2016) The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process. Adv. Appl. Probab. Special Issue (Nick Bingham Festschrift) 48A, 217-233. See here.
• Davis, R.A., Mikosch, T. and Pfaffel, O. (2016) Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series. Stoch. Proc. Appl. 126, 767-799. See here.
• Davis, R.A., Heiny, J., Mikosch, T. and Xie, X. (2016) Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series. Extremes, 19, 517-547. See here.
• Mikosch, T. and Wintenberger, O. (2016) A large deviations approach to limit theory for heavy-tailed time series. Probab. Th. Rel. Fields, 166, 233-269. See here.
• Buraczewski, D., Damek, E. and Mikosch, T. (2016) Stochastic Models with Power-Law Tails. The equation X=AX+B. Springer Book Series ORFIE, New York. See here.
• Janssen, A., Mikosch, T., Rezapour, M. and Xie, X. (2017) The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model. Bernoulli, to appear. See here.
• Heiny, J. and Mikosch, T. (2017) Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: the iid case. Stoch. Proc. Appl., to appear. See here.
• Heiny, J. (2017) Extreme eigenvalues of sample covariance and correlation matrices. PhD Thesis, Department of Mathematics, University of Copenhagen. See here.
• Matsui, M., Mikosch, T. and Samorodnitsky, G. (2017) Distance covariance for stochastic processes. Probab. Math. Statistics, to appear. See here.

• Questions or comments to the contents of this document should be directed to mikosch@math.ku.dk.