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The aim of this paper is to introduce a central limit theorem and an invariance principle for weighted U-statistics based on stationary random fields. Hsing and Wu (2004) in their paper introduced some asymptotic results for weighted U-statistics based on stationary processes. We show that it is possible also to extend their results for weighted $U$-statistics based on stationary random fields.

We consider a sequence of stochastic processes $(X_n\left(𝐭\right),𝐭\in {[0,1]}^m)$ with continuous trajectories and we show conditions for the tightness of the sequence in the Hölder space with a parameter $\gamma$.