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

Dehling, Herold; Taqqu, Murad S.

Ann. Stat. 17, No. 4, 1767-1783 (1989).

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

Keywords:

U-statistics; Hermite polynomials; empirical process; long-range dependence; Donsker’s functional limit theorem; Hermite process; Mises statistics