  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.
