Finite dimensional models for extremes of Gaussian and non-Gaussian processes. (April 2022)

• Record Type: Journal Article
• Authors: Xu, Hui; Grigoriu, Mircea D.
• Is Part Of: Probabilistic engineering mechanics. Volume 68(2022)
• Publication Date: 2022-04
• Subjects: 60 -- 62; Extremes -- Weak convergence -- Almost sure convergence -- Finite dimensional model -- Karhunen–Loève (KL) representation; Engineering -- Statistical methods -- Periodicals; Mechanics, Applied -- Statistical methods -- Periodicals; Probabilities -- Periodicals; Ingénierie -- Méthodes statistiques -- Périodiques; Mécanique appliquée -- Méthodes statistiques -- Périodiques; Probabilités -- Périodiques; 620.100727
• DOI: 10.1016/j.probengmech.2022.103199
• Languages: English
• ISSNs: 0266-8920

Abstract

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 .