## The Annals of Applied Probability

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 25 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1069786514

Digital Object Identifier
doi:10.1214/aoap/1069786514

Mathematical Reviews number (MathSciNet)
MR2023892

Zentralblatt MATH identifier
1038.60040

#### Citation

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. https://projecteuclid.org/euclid.aoap/1069786514

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