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

Functional central limit theorem and Marcinkiewicz strong law of large numbers for Hilbert-valued U-statistics of absolutely regular data