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On random almost periodic trigonometric polynomials and applications to ergodic theory. (English) Zbl 1100.37005

The paper studies random sums of the form \[ \sum^n_{k=1}X_k \exp\bigl[i\langle \lambda_k,t\rangle\bigr], \] where \(\lambda_k\), \(t\in \mathbb{R}^s\) for some \(s\) and where \(X_k\) are random, independent or bounded satisfying some mixing condition. The main result gives uniform estimates on compact sets of these expressions generalizing results of M. Weber [Math. Inequal. Appl. 3, 443–457 (2000; Zbl 0971.60036)] and A. Fan and D. Schneider [Ann. Inst. Henri Poincaré, Probab. Stat. 39, 193–216 (2003; Zbl 1019.42004)]. These estimates are used to derive theorems on uniform convergence, random ergodic theorems for commuting measure preserving transformations and the Wiener-Wintner property [cf. I. Assani, Ergodic Theory Dyn. Syst. 23, 1637–1654 (2003; Zbl 1128.37300)].

MSC:

37A50 Dynamical systems and their relations with probability theory and stochastic processes
60F15 Strong limit theorems
47A35 Ergodic theory of linear operators
42A05 Trigonometric polynomials, inequalities, extremal problems
37A30 Ergodic theorems, spectral theory, Markov operators
37A05 Dynamical aspects of measure-preserving transformations
42A61 Probabilistic methods for one variable harmonic analysis
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