Extremes of Random Processes by Finite Dimensional (FD) Models

Mircea D. Grigoriu
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Mircea D. Grigoriu: Cornell University

Chapter Chapter 5 in Numerical Methods for Extreme Responses of Dynamical Systems, 2025, pp 147-214 from Springer

Abstract: Abstract Conditions are established under which the distributions of extremes of random processes can be approximated by those of extremes of their FD models. The conditions are crucial since most stochastic problems encountered applications have to be solved numerically, numerical methods can only solve finite dimensional problems, i.e., problems involving finite sets of random variables, and continuous time processes are uncountable families of random variables. Four types of FD models are constructed and analyzed. They have the same functional form, i.e., sums of four different deterministic basis functions with random coefficients. The construction and performance of these models is illustrated by numerous examples. The illustration includes the construction of FD models from data by using approximations of the Rosenblatt transformation.

Keywords: Finite dimensional (FD) models; Hat; Sinc; Eigenvalue; Trigonometric basis functions; Polynomial chaos and related approximations; Rosenblatt’s transformation; Weak convergence of FD models (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-75023-6_5

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DOI: 10.1007/978-3-031-75023-6_5

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