High-dimensional extremes and random matrix structures

The study of eigenvalues and eigenvectors of large random matrices is an active and competitive research area with applications ranging from high-dimensional data analysis and machine learning to statistical physics.

The main purpose of this proposal is to develop a comprehensive understanding between the extremal entries and the spectral properties of random matrices.

It is well-known that the largest eigenvalues of sample covariance and Wigner matrices are essentially either Tracy-Widom or Fréchet distributed in the large-dimensional limit.

We will prove a conjecture that unifies both cases.Based on recent probabilistic and combinatorial advances we extend the concept of self-normalization to various dependence measures of multivariate distributions which will yield distribution-free tests that are user-friendly and applicable to a wide variety of problems, for example, in finance and genetical engineering.

In the big data era, these research fields are topical and underwent significant developments in the last decade.

Our results will provide the necessary theoretical foundation for new statistical techniques for the analysis of high-dimensional data.

Research groups in the financial and actuarial sector might incorporate our findings to improve their predictions and hedging algorithms. This proposal contains 8 research projects of which 3 are suitable as starting projects for a PhD student.