ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, April 2005, Vol. 9, pp. 98-115
DOI: 10.1051/ps:2005004

Limit theorems for U-statistics indexed by a one dimensional random walk

Nadine Guillotin-Plantard and Véronique Ladret

Université Claude Bernard, Lyon 1, 50 av. Tony-Garnier, 69366 Lyon Cedex 07, France; nadine.guillotin@univ-lyon1.fr; veronique.ladret@univ-lyon1.fr

(Received August 15, 2004.)

Abstract
Let $(S_{n})_{n\geq 0}$ be a $\mathbb Z$ -random walk and $(\xi_{x})_{x\in \mathbb Z}$ be a sequence of independent and identically distributed $\mathbb R$ -valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on $\mathbb R^2$ with values in $\mathbb R$ . We study the weak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$ , with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by

$\begin{displaymath}\sum_{i,j=0}^{[nt]}h(\xi_},\xi_}), t\in[0,1]. \end{displaymath}$

Statistical applications are presented, in particular we prove a strong law of large numbers for U-statistics indexed by a one-dimensional random walk using a result of [1].

Mathematics Subject Classification. 60F05, 60J15.

Key words: Random walk, random scenery, U-statistics, functional limit theorem.

© EDP Sciences, SMAI 2005