Simulation of Nonstationary Gaussian Processes by Random Trigonometric Polynomials

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Abstract
A family of trigonometric polynomials Xn(t), t≥0, of order n=1,2,… with correlated Gaussian coefficients is used to approximate a general nonstationary Gaussian process X(t) on an arbitrary bounded interval (0,T). The probabilistic characterization of the Gaussian coefficients of Xn(t) can be obtained from the coefficients of the Fourier expansion of the covariance function of X(t) on (0,T)×(0,T). It is shown that the polynomials Xn(t) can match the finite dimensional distributions of X(t) on (0,T) to any degree of accuracy provided that the order n is sufficiently large. An algorithm is developed for generating realizations of X(t), based on the approximating trigonometric polynomials Xn(t). The algorithm involves two phases. First, samples of the Gaussian coefficients of Xn(t) have to be generated. Second, these samples can be used to calculate realizations of Xn(t). The proposed simulation algorithm is simple, efficient and general. An example is presented to demonstrate the proposed simulation method.

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