On random almost periodic trigonometric polynomials and applications to ergodic theory

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 $\{{\lambda }_n^{(i)}:1\le i\le s\}$ 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.