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November 2003 On convergence of the uniform norms for Gaussian processes and linear approximation problems
J. Hüsler, V. Piterbarg, O. Seleznjev
Ann. Appl. Probab. 13(4): 1615-1653 (November 2003). DOI: 10.1214/aoap/1069786514

Abstract

We consider the large values and the mean of the uniform norms for a sequence of Gaussian processes with continuous sample paths. The convergence of the normalized uniform norm to the standard Gumbel (or double exponential) law is derived for distributions and means. The results are obtained from the Poisson convergence of the associated point process of exceedances for a general class of Gaussian processes. As an application we study the piecewise linear interpolation of Gaussian processes whose local behavior is like fractional (integrated fractional) Brownian motion (or with locally stationary increments). The overall interpolation performance for the random process is measured by the $p$th moment of the approximation error in the uniform norm. The problem of constructing the optimal sets of observation locations (or interpolation knots) is done asymptotically, namely, when the number of observations tends to infinity. The developed limit technique for a sequence of Gaussian nonstationary processes can be applied to analysis of various linear approximation methods.

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J. Hüsler. V. Piterbarg. O. Seleznjev. "On convergence of the uniform norms for Gaussian processes and linear approximation problems." Ann. Appl. Probab. 13 (4) 1615 - 1653, November 2003. https://doi.org/10.1214/aoap/1069786514

Information

Published: November 2003
First available in Project Euclid: 25 November 2003

zbMATH: 1038.60040
MathSciNet: MR2023892
Digital Object Identifier: 10.1214/aoap/1069786514

Subjects:
Primary: 60F05 , 60G15 , 60G70

Keywords: $p$th moment convergence , fractional Brownian motion , Maxima of Gaussian processes , piecewise linear approximation , uniform norm

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.13 • No. 4 • November 2003
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