## The empirical process of some long-range dependent sequences with an application to U-statistics.(English)Zbl 0696.60032

Let $$(X_ j)^{\infty}_{j=1}$$ be a stationary, centered, Gaussian chain with strictly positive covariance $$r(k)={\mathcal O}(K^{-D})$$ for large $$k\in \underline N$$ and some $$D\in (0,1)$$. For any measurable function G(.) consider the chain $$(G(X_ j))$$, the d.f. F(.) of the r.v. $$G(X_ 1)$$, and the empirical process-field $e_ N(.,.)=\{d_ N^{- 1}[Nt](F_{[Nt]}(x)-F(x));\quad -\infty \leq x\leq \infty,\quad 0\leq t\leq 1\}$ as a random element in the space D([-$$\infty,\infty]\times [0,1])$$ with the uniform norm. The main theorem in the work generalizes the classical Donsker’s functional limit theorem: for appropriate renormalizations $$d_ N$$ the process $$e_ N$$ converges weakly to the so called Hermite process. As consequences the Hermitian limits are obtained for U-statistics and von Mises statistics using their integral representations through $$e_ N$$, as well.
Reviewer: E.I.Trofimov

### MSC:

 60F17 Functional limit theorems; invariance principles 62G30 Order statistics; empirical distribution functions
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