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On random almost periodic trigonometric polynomials and applications to ergodic theory

Guy Cohen·Christophe Cuny
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Abstract

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(λ_k^{(1)}t_1+...+λ_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{λ_n^{(i)}:1\leq i\leq s\}$ 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. 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.

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Guy Cohen, Christophe Cuny. 2006-02-24. On random almost periodic trigonometric polynomials and applications to ergodic theory. https://doi.org/10.1214/009117905000000459

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