  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.
