On random almost periodic trigonometric polynomials and applications to ergodic theory

Details On random almost periodic trigonometric polynomials and applications to ergodic theory On random almost periodic trigonometric polynomials and applications to ergodic theory

2006-02-24

arxiv.org/abs/math/0602543v1

Annals of Probability 2006, Vol. 34, No. 1, 39-79

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000459 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000459

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered $\{X_n\}$ or bounded $\{X_n\}$ 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. Poincar\'{e} 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.

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