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

Nadine Guillotin-Plantard[1]; Véronique Ladret

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

ESAIM: Probability and Statistics (2005)

  • Volume: 9, page 98-115
  • ISSN: 1292-8100

Abstract

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Let ( S n ) n ≥ 0 be a ℤ -random walk and ( ξ x ) x ∈ 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 ∈ ℕ , with values in D [ 0 , 1 ] the set of right continuous real-valued functions with left limits, defined by ∑ i , j = 0 [ n t ] 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].

How to cite

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Guillotin-Plantard, Nadine, and Ladret, Véronique. "Limit theorems for U-statistics indexed by a one dimensional random walk." ESAIM: Probability and Statistics 9 (2005): 98-115. <http://eudml.org/doc/245979>.

@article{Guillotin2005,
abstract = {Let $(S_\{n\})_\{n\ge 0\}$ 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 $\{\mathcal \{U\}\}_\{n\}, n\in \mathbb \{N\}$, with values in $D[0,1]$ the set of right continuous real-valued functions with left limits, defined by\[ \hspace*\{-56.9055pt\}\sum \_\{i,j=0\}^\{[nt]\}h(\xi \_\{S\_\{i\}\},\xi \_\{S\_\{j\}\}), t\in [0,1]. \]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].},
affiliation = {Université Claude Bernard- Lyon 1, LaPCS - 50 avenue Tony Garnier, 69366 Lyon Cedex 07 (France)},
author = {Guillotin-Plantard, Nadine, Ladret, Véronique},
journal = {ESAIM: Probability and Statistics},
keywords = {random walk; random scenery; $U$-statistics; functional limit theorem; Random walk; U-statistics},
language = {eng},
pages = {98-115},
publisher = {EDP-Sciences},
title = {Limit theorems for U-statistics indexed by a one dimensional random walk},
url = {http://eudml.org/doc/245979},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Guillotin-Plantard, Nadine
AU - Ladret, Véronique
TI - Limit theorems for U-statistics indexed by a one dimensional random walk
JO - ESAIM: Probability and Statistics
PY - 2005
PB - EDP-Sciences
VL - 9
SP - 98
EP - 115
AB - Let $(S_{n})_{n\ge 0}$ 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 ${\mathcal {U}}_{n}, n\in \mathbb {N}$, with values in $D[0,1]$ the set of right continuous real-valued functions with left limits, defined by\[ \hspace*{-56.9055pt}\sum _{i,j=0}^{[nt]}h(\xi _{S_{i}},\xi _{S_{j}}), t\in [0,1]. \]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].
LA - eng
KW - random walk; random scenery; $U$-statistics; functional limit theorem; Random walk; U-statistics
UR - http://eudml.org/doc/245979
ER -

References

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