On random almost periodic trigonometric polynomials and applications to ergodic theory

We study random exponential sums of the form ∑ _k=1 ⁿ X _k exp{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 ≤ i ≤ 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.