We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences and to a generic class of ultrametric Gaussian processes.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-3,
author = {Michel J. G. Weber},
title = {On small deviations of Gaussian processes using majorizing measures},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {41-59},
zbl = {1286.60034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-3}
}
Michel J. G. Weber. On small deviations of Gaussian processes using majorizing measures. Colloquium Mathematicae, Tome 126 (2012) pp. 41-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-3/