Random Matrix Theory with Applications in High Dimensional Statistics and Finance

Nestor Parolya (EUV Frankfurt Oder)
Monday, May 27, 2013 - 2:00pm
Spandauer Strasse 1, Room 23

The random matrix theory (RMT) is originated from the multivariate statistics, nuclear physics and quantum mechanics under the strong impetus of Dyson, Gaudin, Mehta, Wigner and others in the 1960's and 1970's. In particular, in 1967 Marchenko and Pastur derive the well-known equation for the limiting spectral measure for the large dimensional sample covariance matrix. Besides its long history and recent tremendous advances, RMT has emerged as an extremely powerful tool for a variety of applications, especially in statistical signal processing, wireless communications, statistical finance and bioinformatics. In this talk we will give an overview of the asymptotic probability theory of the random matrices and apply the RMT on the large dimensional statistics and finance. Our particular interest covers the case when both the dimension $p$ and the sample size $n$ tend to infinity so that their ratio $p/n$ tends to a positive constant $c$. The traditional estimators usually fail under this large dimensional asymptotics. In this case, we show how to construct the suitable estimators for the large dimensional covariance matrix and its inverse using the RMT. Moreover, the feasible estimators for the optimal portfolio weights and their characteristics are constructed. We show that our estimators obey the rare optimality property, namely they possess almost surely the smallest quadratic loss (Frobenius loss, out-of-sample variance etc.) asymptotically. We do not impose any particular distribution or structure on the asset returns. For our theoretical findings only the existence of the 4th moments and some regularity conditions are needed.