Limit theorems for U-statistics indexed by a one dimensional random walk
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 98-115.

Let (S n ) n0 be a -random walk and (ξ x ) x be a sequence of independent and identically distributed -valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on 2 with values in . We study the weak convergence of the sequence 𝒰 n ,n, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by

i,j=0 [nt] h(ξ S i ,ξ S j ),t[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].

DOI : 10.1051/ps:2005004
Classification : 60F05, 60J15
Mots-clés : random walk, random scenery, $U$-statistics, functional limit theorem
Guillotin-Plantard, Nadine 1 ; Ladret, Véronique 

1 Université Claude Bernard- Lyon 1, LaPCS - 50 avenue Tony Garnier, 69366 Lyon Cedex 07 (France)
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Guillotin-Plantard, Nadine; Ladret, Véronique. Limit theorems for U-statistics indexed by a one dimensional random walk. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 98-115. doi : 10.1051/ps:2005004. http://archive.numdam.org/articles/10.1051/ps:2005004/

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