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.
Publié le : 2003-11-14
Classification:
Maxima of Gaussian processes,
uniform norm,
$p$th moment convergence,
piecewise linear approximation,
fractional Brownian motion,
60G70,
60G15,
60F05
@article{1069786514,
author = {H\"usler, J. and Piterbarg, V. and Seleznjev, O.},
title = {On convergence of the uniform norms for Gaussian processes and linear approximation problems},
journal = {Ann. Appl. Probab.},
volume = {13},
number = {1},
year = {2003},
pages = { 1615-1653},
language = {en},
url = {http://dml.mathdoc.fr/item/1069786514}
}
Hüsler, J.; Piterbarg, V.; Seleznjev, O. On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab., Tome 13 (2003) no. 1, pp. 1615-1653. http://gdmltest.u-ga.fr/item/1069786514/