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January 2006 On random almost periodic trigonometric polynomials and applications to ergodic theory
Guy Cohen, Christophe Cuny
Ann. Probab. 34(1): 39-79 (January 2006). DOI: 10.1214/009117905000000459


We study random exponential sums of the form ∑k=1nXkexp{i(λk(1)t1+⋯+λk(s)ts)}, where {Xn} is a sequence of random variables and {λn(i):1≤is} are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered {Xn} or bounded {Xn} 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.


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Guy Cohen. Christophe Cuny. "On random almost periodic trigonometric polynomials and applications to ergodic theory." Ann. Probab. 34 (1) 39 - 79, January 2006.


Published: January 2006
First available in Project Euclid: 17 February 2006

zbMATH: 1100.37005
MathSciNet: MR2206342
Digital Object Identifier: 10.1214/009117905000000459

Primary: 37A50 , 60F15
Secondary: 42A05 , 47A35

Keywords: Almost everywhere convergence , Banach-valued random variables , Maximal inequalities , Moment inequalities , random Fourier series

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 1 • January 2006
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