Abstract

Though the area of random matrices has grown rapidly within physics and mathematics, it is interesting to note that its first appearance was in Statistics. In the last couple of decades, the theory and applications of random matrices has grown enormously across many different scientific disciplines, for example in particle physics, operator theory, wireless communications, probability, and non-commutative probability. Though large dimensional random matrices (LDRM) has been gaining some attention in high dimensional statistics, statisticians by and large have remained somewhat unfamiliar with various aspects of LDRM.

The goal of this lecture is to quickly introduce some results for sequences of LRDM, specially from the point of view of the behaviour of the limit spectrum of one or more sequences of random matrices and point out some potential applications in statistics.
In particular, I hope to discuss some symmetric high dimensional random matrices (such as the Wigner matrix, the sample covariance and symmetrised auto-covariance matrices), and some non-symmetric high dimensional matrices (such as the i.i.d. matrix and the sample autocovariance matrices). I will also point out some potential applications of the results on autocovariance matrices to inference problems in high dimensional time series.