Monte Carlo estimates of extremes of stationary/nonstationary Gaussian processes

Abstract Finite-dimensional (FD) models X d ⁢ ( t ) X_{d}(t) , i.e., deterministic functions of time and finite sets of 𝑑 random variables, are constructed for stationary and nonstationary Gaussian processes X ⁢ ( t ) X(t) with continuous samples defined on a bounded time interval [ 0 , τ ] [0,\tau] . The basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of X ⁢ ( t ) X(t) and of trigonometric functions. Numerical illustrations are presented for a stationary Gaussian process X ⁢ ( t ) X(t) with exponential correlation function and a nonstationary version of this process obtained by time distortion. It was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.