Simulation of Nonstationary Gaussian Processes by Random Trigonometric Polynomials

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