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 : https://doi.org/10.1051/ps:2005004
Classification : 60F05,  60J15
Mots clés : random walk, random scenery, U-statistics, functional limit theorem
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     author = {Guillotin-Plantard, Nadine and Ladret, V\'eronique},
     title = {Limit theorems for {U-statistics} indexed by a one dimensional random walk},
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     pages = {98--115},
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     url = {http://www.numdam.org/articles/10.1051/ps:2005004/}
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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://www.numdam.org/articles/10.1051/ps:2005004/

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