Distribution approximations for nonlinear functionals of weakly correlated random processes (Q1355121)

Summary: Nonlinear functionals of weakly correlated processes with correlation length \(\varepsilon> 0\) are investigated. Expansions of moments and distribution densities of nonlinear functionals with respect to \(\varepsilon\) up to terms of order \(o(\varepsilon)\) are considered. For the case of a single nonlinear functional a shorter proof than in [the first author, ``Stochastic equations of mathematical physics'' (1990; Zbl 0715.60061)] is given. The results are applied to eigenvalues of random matrices which are obtained by application of the Ritz method to random differential operators. Using the expansion formulas as to \(\varepsilon\) approximations of the density functions of the matrix eigenvalues can be found. In addition to [the first author and \textit{W. Purkert}, ``Random eigenvalue problems'' (1984; Zbl 0585.60062)] not only first-order approximations (exact up to terms of order \(O(\varepsilon)\)) but also second-order approximations (exact up to terms of order \(o(\varepsilon)\)) are investigated. These approximations are compared with estimations from Monte-Carlo simulation.