In this paper, we investigate the functional central limit theorem and the Marcinkiewicz strong law of large numbers for U-statistics having absolutely regular data and taking value in a separable Hilbert space. The novelty of our approach consists in using coupling in order to formulate a deviation inequality for original $U$-statistic, where the upper bound involves the mixing coefficient and the tail of several U-statistics of i.i.d. data. The presented results improve the known results in several directions: the case of metric space valued data is considered as well as Hilbert space valued, and the mixing rates are less restrictive in a wide range of parameters.

翻译：本文研究了取值于可分Hilbert空间、具有绝对正则数据的U-统计量的泛函中心极限定理与Marcinkiewicz强大数定律。我们方法的新颖之处在于利用耦合技术推导原始U-统计量的偏差不等式，其上界涉及混合系数与若干独立同分布数据U-统计量的尾部特征。所得结果在以下方面改进了已有结论：既考虑了度量空间值数据情形，也涵盖了Hilbert空间值情形，且在较大参数范围内混合速率限制更为宽松。