Simulation of Stationary Gaussian Processes in [0, 1]d

Abstract

A method for simulating a stationary Gaussian process on a fine rectangular grid in [0, 1]^d ⊂ℝ^d is described. It is assumed that the process is stationary with respect to translations of ℝ^d, but the method does not require the process to be isotropic. As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform. However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied. It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on ℝ^d, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on ℝ^d, is strictly positive. The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer. An approximate procedure is also proposed to cover cases where it is not feasible to apply the procedure in its exact form.