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.

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 | 1 citation dans Numdam

DOI : 10.1051/ps:2005004

Classification : 60F05, 60J15
Mots-clés : 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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     publisher = {EDP-Sciences},
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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. https://www.numdam.org/articles/10.1051/ps:2005004/

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Alvarez-Andrade, Sergio; Bouzebda, Salim Cramér’s type results for some bootstrapped U-statistics, Statistical Papers, Volume 61 (2020) no. 4, p. 1685 | DOI:10.1007/s00362-018-0999-8
Franke, Brice; Pène, Françoise; Wendler, Martin Convergence of $U$ -statistics indexed by a random walk to stochastic integrals of a Lévy sheet, Bernoulli, Volume 23 (2017) no. 1 | DOI:10.3150/15-bej745
Franke, Brice; Pène, Françoise; Wendler, Martin Stable limit theorem for $U$ -statistic processes indexed by a random walk, Electronic Communications in Probability, Volume 22 (2017) no. none | DOI:10.1214/16-ecp4173
Wendler, Martin The sequential empirical process of a random walk in random scenery, Stochastic Processes and their Applications, Volume 126 (2016) no. 9, p. 2787 | DOI:10.1016/j.spa.2016.03.002
References, Dynamic Random Walks (2006), p. 249 | DOI:10.1016/b978-044452735-6/50054-3

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