随机概周期三角多项式及其在遍历论中的应用
On random almost periodic trigonometric polynomials and applications to ergodic theory
摘要：
研究了$\sum_{k=1}^nX_k\times\ex p\{i(λ_k^{(1)}t_1+...+λ_k^{(s)}t_s)\}$型随机指数和，其中$\{X_n\}$是随机变量序列，$\{λ_n^{(i)}:1\leq i\leq s\}$是实数序列.对于独立中心的$\{X_n\}$或有界的$\{X_n\}$满足一定的混合条件，我们得到了这些和的一致估计（在紧集上）.这些结果推广了Weber [Math. Inequal. 3（2000）443- 457]和Fan和Schneider [Ann.Inst.H.1999]。庞加莱概率中央集权主义者39（2003）193- 216]。作为应用，我们得到了这些和在紧集上一致收敛性的条件.我们还得到了概率空间的多个交换保测点变换的随机遍历定理。最后，我们展示了我们的一些结果如何允许导出Wiener-Wintner性质（由Assani [遍历论Dynam. Systems 23（2003）1637- 1654]），用于某些动力系统上的某些函数。
英文摘要：
We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(λ_k^{(1)}t_1+...+λ_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{λ_n^{(i)}:1\leq i\leq s\}$ 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.
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