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

Xiaolei Xie (since October 1, 2014)

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:

Presentation of the results of the project

Publications related to the project

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