Asymptotics of random matrices and matrix valued processes

Abstract: This thesis contains three parts. In the first two papers we consider spectral properties of symmetric matrices with elements consisting of independent Ornstein Uhlenbeck processes. The eigenvalues behave as a particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force. The empirical measure is known to converge weakly in a space of continuous measure valued functions.In the first paper we let the empirical measure act on polynomial functions and on functions of the type exp(cx2). These functionals are shown to converge under suitable conditions in the space of continuous real valued functions on [0,T].We prove in the second paper that under suitable conditions the fluctuations of the empirical measure valued process around the limiting measure valued process, appropriately scaled, converge weakly to a Gaussian distribution valued process.In the last paper we prove limit theorems for functionals of random matrices. Assuming that Yn is an n x n Wigner matrix we construct a new class of random matrices by letting Xn=Σn+ΣimBinYnCin where Σn,Bin,Cin are deterministic. We study the sequence of trace functionals 1ntr(An(Xn+zIn)-1) and prove convergence in probability to a limit c for which a representation formula is given.