Speaker: Jeff Yao

Title: Some fundamental results in random matrix theory with applications to information theory and high-dimensional statistics

Abstract: This talk is aimed at a general survey of the theory of random matrices (RMT) with connections to information theory and high-dimensional statistics. RMT traces back to the development of quantum mechanics in the 1940s and early 1950s with the discovery of the celebrated semi-circle law by Eugene P. Wigner. The theory has been since then extended to many different classes of large matrices, such as sample covariance matrices, Toeplitz matrices, Markov matrices or arbitrary rectangular matrices to name a few.  In the early 1980s, major contributions in RMT relate to the existence of a limiting distribution for the eigenvalues of a large random matrix. In recent years, research on RMT has turned toward second-order limiting theorems, such as the central limit theorem for linear spectral statistics, the limiting distributions of spectral spacings, and of extreme eigenvalues. These fast developments are much motivated by the intersection of RMT with a number of very different areas such as quantum mechanics, wireless communications, number theory or high-dimensional statistics.

   In this talk, I will describe a few selected important results from RMT which are meaningful to the area of information theory and of high-dimensional statistics. As far as possible, hints on mathematical tools leading to these results will be discussed. When coming to the applications to information theory and high-dimensional statistics, I will try to show the fundamental benefits obtained with RMT with respect to classical mathematical tools.