Random Vectors and Processes

Mircea D. Grigoriu
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Mircea D. Grigoriu: Cornell University

Chapter Chapter 3 in Numerical Methods for Extreme Responses of Dynamical Systems, 2025, pp 57-116 from Springer

Abstract: Abstract Random processes are viewed as continuous versions of random vectors to facilitate the understanding of the mean/correlation functions, the finite dimensional distributions, and other descriptors for real- and vector-valued processes. It is shown that weakly stationary and nonstationary processes can be represented by sums of sines and cosines with random amplitudes (spectral representation) or sums of eigenfunctions of correlation functions with random amplitudes (Kahunen-Lo‘eve expansion). The chapter also discusses processes encountered frequently in applications, dedicates an entire section to the Brownian motion process that is used extensively in the book, and examines extremes of FD models.

Keywords: Brownian motion; Finite dimensional (FD) models; Gauss; Translation and Markov processes; Kahunen-Loève expansion; Spectral representation; Weakly stationary processes (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/978-3-031-75023-6_3

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