Self averaging of normalized spectral functions of some product of independent random matrices of growing dimension

Abstract

The problem of the spectral analysis of random matrizant (the product of random matrices), which is the solution of a recurrent system of equations with random coefficients, or the system of stochastic linear differential equations of growing dimension is considered. The growing dimension means that the dimension of matrices and the number of matrices have the same order and both (dimension and number of matrices) tend to infinity. In this paper we give new method of deriving self averaging property for the V.I.C.T.O.R.I.A.-transform of normalized spectral functions (n.s.f.) of random matrizant or the product of independent random matrices. We apply the REFORM method for normalized spectral functions of this matrizant, where random matrices belong to the domain of attraction of the Strong Circular Law.