On random almost periodic trigonometric polynomials and applications to ergodic theory

• 成果类型: Article
• 署名作者: Cohen, G; Cuny, C
• 署名单位: Ben-Gurion University of the Negev
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000459
• 发表日期: 2006
• 页码: 39-79
• 关键词: series; LAWS

摘要

We study random exponential sums of the form Sigma(n)(k=1) X-k exp {i (lambda((1))(k) t(1) +(...)+ lambda((s))(k)t(s))}, where {X-n} is a sequence of random variables and {lambda((i))(n) : 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. Poincare 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.