Distribution Approximations for Nonlinear Functionals of Weakly Correlated Random Processes

Abstract

In this paper nonlinear functionals of weakly correlated processes with correlation length $ϵ>0$ are investigated. Expansions of moments and distribution densities of nonlinear functionals with respect to $ϵ$ up to terms of order $o(ϵ)$ are considered. For the case of a single nonlinear functional a shorter proof than in [8] is given. The results are applied to cigenvalues of random matrices which are obtained by application of the Ritz method to random differential operators. Using the expansion formulas as to e approximations of the density functions of the matrix cigenvalues can be found. In addition to [7] not only first order approximations (exact up to terms of order $O(ϵ)$) but also second order approximations (exact up to terms of order $o(ϵ)$) are investigated. These approximations are compared with estimations from Monte-Carlo simulation.