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## Asymptotic properties of U-processes under long-range dependence

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Céline Lévy-Leduc
• Function : Author
Valderio A. Reisen
• Function : Author

#### Abstract

Let $(X_i)_{i\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\rho(k)=\PE(X_{1}X_{k+1})$ satisfying: $\rho(0)=1$ and $\rho(k)=k^{-D} L(k)$ where $D$ is in $(0,1)$ and $L$ is slowly varying at infinity. Consider the $U$-process $\{U_n(r),\; r\in I\}$ defined as $U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j\leq n}\1_{\{G(X_i,X_j)\leq r\}}\; ,$ where $I$ is an interval included in $\rset$ and $G$ is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated scale estimator. The limiting distributions are expressed through multiple Wiener-Itô integrals.

### Dates and versions

hal-00442874 , version 1 (23-12-2009)
hal-00442874 , version 2 (03-12-2010)

### Identifiers

• HAL Id : hal-00442874 , version 2
• ARXIV :

### Cite

Céline Lévy-Leduc, Hélène Boistard, Éric Moulines, Murad S. Taqqu, Valderio A. Reisen. Asymptotic properties of U-processes under long-range dependence. 2009. ⟨hal-00442874v2⟩

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