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

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

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## Abstract

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Let (Sn)n≥0 be a $ℤ$-random walk and ${\left({\xi }_{x}\right)}_{x\in ℤ}$ 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 $\sum _{i,j=0}^{\left[nt\right]}h\left({\xi }_{{S}_{i}},{\xi }_{{S}_{j}}\right),t\in \left[0,1\right].$ 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 (2010): 98-115. <http://eudml.org/doc/104343>.

@article{Guillotin2010,
abstract = { Let (Sn)n≥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 $\{\cal U\}_\{n\}, n\in \mathbb N$, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by $\sum\_\{i,j=0\}^\{[nt]\}h(\xi\_\{S\_\{i\}\},\xi\_\{S\_\{j\}\}), t\in[0,1].$ 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]. },
journal = {ESAIM: Probability and Statistics},
keywords = {Random walk; random scenery; U-statistics; functional limit theorem.; U-statistics; functional limit theorem},
language = {eng},
month = {3},
pages = {98-115},
publisher = {EDP Sciences},
title = {Limit theorems for U-statistics indexed by a one dimensional random walk},
url = {http://eudml.org/doc/104343},
volume = {9},
year = {2010},
}

TY - JOUR
TI - Limit theorems for U-statistics indexed by a one dimensional random walk
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 98
EP - 115
AB - Let (Sn)n≥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 ${\cal U}_{n}, n\in \mathbb N$, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by $\sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1].$ 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.; U-statistics; functional limit theorem
UR - http://eudml.org/doc/104343
ER -

## References

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