On convergence of the uniform norms for Gaussian processes and linear approximation problems. (English) Zbl 1038.60040

This paper considers 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 this paper studies 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 pth 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 non-stationary processes can be applied to the analysis of various linear approximation methods.


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