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

Guillotin-Plantard, Nadine; Ladret, Véronique

doi:10.1051/ps:2005004

Guillotin-Plantard, Nadine ¹ ; Ladret, Véronique

¹ Université Claude Bernard- Lyon 1, LaPCS - 50 avenue Tony Garnier, 69366 Lyon Cedex 07 (France)

ESAIM: Probability and Statistics, Tome 9 (2005), pp. 98-115

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Résumé

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

\sum_{i, j = 0}^{[n t]} h (ξ_{S_{i}}, ξ_{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].

MR Zbl

DOI : 10.1051/ps:2005004

Classification : 60F05, 60J15
Keywords: random walk, random scenery, $U$-statistics, functional limit theorem

Affiliations des auteurs :

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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     author = {Guillotin-Plantard, Nadine and Ladret, V\'eronique},
     title = {Limit theorems for {U-statistics} indexed by a one dimensional random walk},
     journal = {ESAIM: Probability and Statistics},
     pages = {98--115},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2005},
     doi = {10.1051/ps:2005004},
     mrnumber = {2148962},
     zbl = {1136.60316},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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

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