Finite dimensional models for extremes of Gaussian and non-Gaussian processes. (April 2022) - Xu, Hui; Grigoriu, Mircea D.
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 X d ( 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 X d ( 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 X d ( t ) and that the discrepancy between samples and extremes of these processes decreases with d .