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

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu and V. A. Reisen.

Let $(X_i)_{i\geq1}$ be a stationary mean-zero Gaussian process with covariances $\rho(k) = E(X_1X_{k+1})$ 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
$U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j \leq n}\mathbb{1}_{\{G(X_i,X_j\leq r\}},$
where I is an interval included in $\mathbb{R}$ 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, in the long-range dependence setting, new properties of many well-known estimators such as the Hodges- Lehmann estimator, which is a well-known robust location estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated robust scale estimator. These robust estimators are shown to have the same asymptotic distribution as the classical location and scale estimators. The limiting distributions are expressed through multiple Wiener-Itô integrals.

# Central limit theorem for multiple integrals with respect to the empirical process

Joint work with Eustasio del Barrio. Article publicated in Statistics and Probability Letters, vol. 79(2), p. 188-195, January 2009.
$J_{n,m}(h)=\int'h(x_1, \dots, x_m)d\mathbb{G}_n(x_1)\dots \mathbb{G}_n(x_m),$
where h is a symmetric real-valued square integrable function of m variables. $X_1, \dots, X_n$ is a P-distributed i.i.d. sample, and $\mathbb{P}_n=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$ and $\mathbb{G}_n=\sqrt{n}(\mathbb{P}_n-P)$ are respectively the associated empirical measure and the empirical process. $\int'$ is the integral where the integration on the diagonal has been omitted. We include the case of non degenerate kernels with respect to the underlying distribution. Our results are related to earlier results on U-statistics. We introduce a stochastic integral with respect to the Brownian bridge which allows us to express the limit in a unified way in the degenerate and non degenerate cases. Using the multiple integral with respect to the empirical process has an advantage with respect to using U-statistics: the Central Limit Theorem we obtain is simpler. It does not involve the degeneracy of the kernel and the limit is expressed in a precise way.