On random almost periodic trigonometric polynomials and applications to ergodic theory

Guy Cohen, Christophe Cuny

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)39-79
Number of pages41
JournalAnnals of Probability
Volume34
Issue number1
DOIs
StatePublished - 1 Jan 2006

Keywords

  • Almost everywhere convergence
  • Banach-valued random variables
  • Maximal inequalities
  • Moment inequalities
  • Random Fourier series

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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