Statistical applications are presented, in particular we prove a strong law of large numbersfor U-statistics indexed by a one-dimensional random walk using a result of [1]." name="description" /> ESAIM: Probability and Statistics

ESAIM: Probability and Statistics

Table of Contents - November 2005 - Volume 9  

Research Article

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

Guillotin-Plantard, Nadinea1 and Ladret, Véroniquea1

a1 Université Claude Bernard, Lyon 1, 50 av. Tony-Garnier, 69366 Lyon Cedex 07, France; nadine.guillotin@univ-lyon1.fr; veronique.ladret@univ-lyon1.fr

Abstract

Let (Sn)n≥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 \[ \sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1]. \] 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].

(Received August 15 2004)

(Online publication November 15 2005)

Key Words:

  • Random walk;
  • random scenery;
  • U-statistics;
  • functional limit theorem.

Mathematics Subject Classification:

  • 60F05;
  • 60J15
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