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 -
References
top- J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill and B. Weiss, Strong laws for L- and U-statistics. Trans. Amer. Math. Soc.348 (1996) 2845–2866.
- P. Billingsley, Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition. A Wiley-Interscience Publication (1999).
- E. Bolthausen, A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab.17 (1989) 108–115.
- E. Boylan, Local times for a class of Markoff processes. Illinois J. Math.8 (1964) 19–39.
- E. Buffet and J.V. Pulé, A model of continuous polymers with random charges. J. Math. Phys.38 (1997) 5143–5152.
- P. Cabus and N. Guillotin-Plantard, Functional limit theorems for U-statistics indexed by a random walk. Stochastic Process. Appl.101 (2002) 143–160.
- F. den Hollander, Mixing properties for random walk in random scenery. Ann. Probab.16 (1988) 1788–1802.
- F. den Hollander, M.S. Keane, J. Serafin and J.E. Steif, Weak bernoullicity of random walk in random scenery. Japan. J. Math. (N.S.)29 (2003) 389–406.
- F. den Hollander and J.E. Steif, Mixing properties of the generalized T,T-1-process. J. Anal. Math.72 (1997) 165–202.
- R.K. Getoor and H. Kesten, Continuity of local times for Markov processes. Comp. Math.24 (1972) 277–303.
- W. Hoeffding, The strong law of large numbers for U-statistics. Univ. N. Carolina, Institue of Stat. Mimeo series302 (1961).
- H. Kesten and F. Spitzer, A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete50 (1979) 5–25.
- A.J. Lee, U-statistics. Theory and practice. Marcel Dekker, Inc., New York (1990).
- M. Maejima, Limit theorems related to a class of operator-self-similar processes. Nagoya Math. J.142 (1996) 161–181.
- S. Martínez and D. Petritis, Thermodynamics of a Brownian bridge polymer model in a random environment. J. Phys. A29 (1996) 1267–1279.
- I. Meilijson, Mixing properties of a class of skew-products. Israel J. Math.19 (1974) 266–270.
- D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, Berlin. Fundamental Principles of Mathematical Sciences 293 (1999).
- R.J. Serfling, Approximation theorems of mathematical statistics. John Wiley & Sons Inc., New York. Wiley Series in Probability and Mathematical Statistics (1980).
- F. Spitzer, Principles of random walks. Springer-Verlag, New York, second edition. Graduate Texts in Mathematics 34 (1976).
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