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December, 1989 The Empirical Process of some Long-Range Dependent Sequences with an Application to $U$-Statistics
Herold Dehling, Murad S. Taqqu
Ann. Statist. 17(4): 1767-1783 (December, 1989). DOI: 10.1214/aos/1176347394

Abstract

Let $(X_j)^\infty_{j = 1}$ be a stationary, mean-zero Gaussian process with covariances $r(k) = EX_{k + 1} X_1$ satisfying $r(0) = 1$ and $r(k) = k^{-D}L(k)$ where $D$ is small and $L$ is slowly varying at infinity. Consider the two-parameter empirical process for $G(X_j),$ $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where $G$ is any measurable function. Noncentral limit theorems are obtained for $F_N(x, t)$ and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and $U$-statistics based on the $G(X_j)$'s. The limiting processes are structurally different from those encountered in the i.i.d. case.

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Herold Dehling. Murad S. Taqqu. "The Empirical Process of some Long-Range Dependent Sequences with an Application to $U$-Statistics." Ann. Statist. 17 (4) 1767 - 1783, December, 1989. https://doi.org/10.1214/aos/1176347394

Information

Published: December, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0696.60032
MathSciNet: MR1026312
Digital Object Identifier: 10.1214/aos/1176347394

Subjects:
Primary: 60F17
Secondary: 62G30

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 4 • December, 1989
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