Applications of Random Matrix Theory for high-dimensional statistics.
In the present era of Big Data, new statistical methods are needed to decipher large dimensional data sets that are now routinely generated in almost all fields – physics, image analysis, genomics, epidemiology, engineering and finance, to quote only a few. It is very natural to try to identify common causes that explain the joint dynamics of a large number of quantities. The primary aim of this thesis is to understand theoretically the so-called curse of dimensionality that describe phenomena which arise in high-dimensional space using Random Matrix Theory. Special care is devoted to the statistics of the eigenvectors of large noisy matrices, which turn out to be crucial for many applications. Moreover, I will present how to build reliable estimators that are consistent with the dimension of the problem. In the case of correlation matrices, the estimator we obtain provide better performance than all previously proposed methods for real-world applications within financial markets.