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

Nadine Guillotin-Plantard; Véronique Ladret

ESAIM: Probability and Statistics (2010)

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

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topGuillotin-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].
},

author = {Guillotin-Plantard, Nadine, Ladret, Véronique},

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

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

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 -

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