An approximate solution to the simultaneous diagonalization of two covariance kernels: applications to second-order stochastic processes

We define the approximate simultaneous orthogonal (ASO) expansions of two second-order stochastic processes from the Rayleigh-Ritz eigenfunctions and prove its convergence. We consider an example that illustrates the implementation of the proposed method and that allows us to assess the accuracy of the approximations achieved with such finite expansions. Finally, we give two specific applications: in the problem of estimating a Gaussian signal in noise and in the Gaussian signal detection problem