The Annals of Applied Probability

On convergence of the uniform norms for Gaussian processes and linear approximation problems

J. Hüsler, V. Piterbarg, and O. Seleznjev

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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.

Article information

Ann. Appl. Probab., Volume 13, Number 4 (2003), 1615-1653.

First available in Project Euclid: 25 November 2003

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Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60G15: Gaussian processes 60F05: Central limit and other weak theorems

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


Hüsler, J.; Piterbarg, V.; Seleznjev, O. On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 (2003), no. 4, 1615--1653. doi:10.1214/aoap/1069786514.

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