A precise link proved by Kuelbs and Li relates the small ball
behavior of a Gaussian measure $\mu$ on a Banach space $E$ with the metric
entropy behavior of $K_\mu$, the unit ball of the reproducing kernel Hilbert
space of $\mu$ in $E$. We remove the main regularity assumption imposed on the
unknown function in the link. This enables the application of tools and results
from functional analysis to small ball problems and leads to small ball
estimates of general algebraic type as well as to new estimates for concrete
Gaussian processes. Moreover, we show that the small ball behavior of a
Gaussian process is also tightly connected with the speed of approximation by
“finite rank” processes.
Publié le : 1999-07-14
Classification:
Gaussian process,
small deviation,
metric entropy,
approximation number,
60G15,
60F99,
47D50,
47G10
@article{1022677459,
author = {Li, Wenbo V. and Linde, Werner},
title = {Approximation, Metric Entropy and Small Ball Estimates for
Gaussian Measures},
journal = {Ann. Probab.},
volume = {27},
number = {1},
year = {1999},
pages = { 1556-1578},
language = {en},
url = {http://dml.mathdoc.fr/item/1022677459}
}
Li, Wenbo V.; Linde, Werner. Approximation, Metric Entropy and Small Ball Estimates for
Gaussian Measures. Ann. Probab., Tome 27 (1999) no. 1, pp. 1556-1578. http://gdmltest.u-ga.fr/item/1022677459/