On random almost periodic trigonometric polynomials and applications to ergodic theory

Abstract

We study random exponential sums of the form ∑_k=1ⁿX_kexp{i(λ_k⁽¹⁾t₁+⋯+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.