Finite dimensional models for extremes of Gaussian and non-Gaussian processes

Abstract

Numerical solutions of stochastic problems involving random processes X(t), which constitutes infinite families of random variables, require to represent these processes by finite dimensional (FD) models Xd(t), i.e., deterministic functions of time depending on finite numbers d of random variables. Most available FD models match the mean, correlation, and other global properties of X(t). They provide useful information to a broad range of problems, but cannot be used to estimate extremes or other sample properties of X(t). We develop FD models Xd(t) for processes X(t) with continuous samples and establish conditions under which these models converge weakly to X(t) in the space of continuous functions as d→∞. These theoretical results are illustrated by numerical examples which show that, under the conditions established in this study, samples and extremes of X(t) can be approximated by samples and extremes of Xd(t) and that the discrepancy between samples and extremes of these processes decreases with d.