Extremes of Nonstationary Harmonizable Processes

Finite dimensional (FD) models Xd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_d$$\end{document}, i.e., deterministic functions of time and finite sets of d random variables, are developed for a class of nonstationary processes X, referred to as harmonizable. The FD models are based on Karhunen-Lo & egrave;ve and spectral representations of X. Conditions are established under which distributions of extremes of X can be approximated by those of extremes of Xd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_d$$\end{document} provided that the stochastic dimension d is sufficiently large. FD models are constructed for monochromatic, Brownian motion and Ornstein-Uhlenbeck processes. Numerical results suggest that their extremes can be used as surrogates for the extremes of these processes in agreement with our theoretical findings.