We study random exponential sums of the form ∑k=1nXkexp{i(λk(1)t1+⋯+λk(s)ts)}, where {Xn} is a sequence of random variables and {λn(i):1≤i≤s} are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered {Xn} or bounded {Xn} satisfying some mixing conditions. These results generalize recent results of Weber [Math. Inequal. Appl. 3 (2000) 443–457] and Fan and Schneider [Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 193–216] in several directions. As applications we derive conditions for uniform convergence of these sums on compact sets. We also obtain random ergodic theorems for finitely many commuting measure-preserving point transformations of a probability space. Finally, we show how some of our results allow to derive the Wiener–Wintner property (introduced by Assani [Ergodic Theory Dynam. Systems 23 (2003) 1637–1654]) for certain functions on certain dynamical systems.
Publié le : 2006-01-14
Classification:
Moment inequalities,
maximal inequalities,
almost everywhere convergence,
random Fourier series,
Banach-valued random variables,
37A50,
60F15,
47A35,
42A05
@article{1140191532,
author = {Cohen, Guy and Cuny, Christophe},
title = {On random almost periodic trigonometric polynomials and applications to ergodic theory},
journal = {Ann. Probab.},
volume = {34},
number = {1},
year = {2006},
pages = { 39-79},
language = {en},
url = {http://dml.mathdoc.fr/item/1140191532}
}
Cohen, Guy; Cuny, Christophe. On random almost periodic trigonometric polynomials and applications to ergodic theory. Ann. Probab., Tome 34 (2006) no. 1, pp. 39-79. http://gdmltest.u-ga.fr/item/1140191532/