Parametric models of nonstationary Gaussian processes

Abstract

Three parametric representations are developed for approximating a general nonstationary Gaussian process X(t). The representations: (1) are based on the Bernstein and other interpolation polynomials, spline functions, and an extension of a sampling theorem for stationary processes; (2) consist of finite sums of specified deterministic functions with random amplitudes depending on X(t); and (3) converge to X(t) as the number of these functions increases. However, their convergence rates differ. Numerical results for a nonstationary Ornstein-Uhlenbeck process show that the interpolation polynomials have the slowest rate of convergence. The parametric representations based on spline functions and the extended sampling theorem have similar convergence rates. The paper also presents methods for generating realizations of X(t) based on the three parametric models of this process.