## 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 language | English |
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Pages (from-to) | 39-79 |

Number of pages | 41 |

Journal | Annals of Probability |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - 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