Limit theorems for U-statistics indexed by a one dimensional random walk
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 98-115.

Let ${\left({S}_{n}\right)}_{n\ge 0}$ 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 $D\left[0,1\right]$ the set of right continuous real-valued functions with left limits, defined by

 $\phantom{\rule{-56.9055pt}{0ex}}\sum _{i,j=0}^{\left[nt\right]}h\left({\xi }_{{S}_{i}},{\xi }_{{S}_{j}}\right),t\in \left[0,1\right].$
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].

DOI : https://doi.org/10.1051/ps:2005004
Classification : 60F05,  60J15
Mots clés : random walk, random scenery, $U$-statistics, functional limit theorem
@article{PS_2005__9__98_0,
title = {Limit theorems for {U-statistics} indexed by a one dimensional random walk},
journal = {ESAIM: Probability and Statistics},
pages = {98--115},
publisher = {EDP-Sciences},
volume = {9},
year = {2005},
doi = {10.1051/ps:2005004},
zbl = {1136.60316},
mrnumber = {2148962},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2005004/}
}
TY  - JOUR
TI  - Limit theorems for U-statistics indexed by a one dimensional random walk
JO  - ESAIM: Probability and Statistics
PY  - 2005
DA  - 2005///
SP  - 98
EP  - 115
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2005004/
UR  - https://zbmath.org/?q=an%3A1136.60316
UR  - https://www.ams.org/mathscinet-getitem?mr=2148962
UR  - https://doi.org/10.1051/ps:2005004
DO  - 10.1051/ps:2005004
LA  - en
ID  - PS_2005__9__98_0
ER  - 
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. http://www.numdam.org/articles/10.1051/ps:2005004/

[1] 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. | Zbl 0863.60032

[2] 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). | MR 1700749 | Zbl 0944.60003

[3] E. Bolthausen, A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989) 108-115. | Zbl 0679.60028

[4] E. Boylan, Local times for a class of Markoff processes. Illinois J. Math. 8 (1964) 19-39. | Zbl 0126.33702

[5] E. Buffet and J.V. Pulé, A model of continuous polymers with random charges. J. Math. Phys. 38 (1997) 5143-5152. | Zbl 0890.60099

[6] P. Cabus and N. Guillotin-Plantard, Functional limit theorems for $U$-statistics indexed by a random walk. Stochastic Process. Appl. 101 (2002) 143-160. | Zbl 1075.60018

[7] F. Den Hollander, Mixing properties for random walk in random scenery. Ann. Probab. 16 (1988) 1788-1802. | Zbl 0651.60108

[8] 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. | Zbl 1049.60041

[9] F. Den Hollander and J.E. Steif, Mixing properties of the generalized $T,{T}^{-1}$-process. J. Anal. Math. 72 (1997) 165-202. | Zbl 0898.60070

[10] R.K. Getoor and H. Kesten, Continuity of local times for Markov processes. Comp. Math. 24 (1972) 277-303. | Numdam | Zbl 0293.60069

[11] W. Hoeffding, The strong law of large numbers for $U$-statistics. Univ. N. Carolina, Institue of Stat. Mimeo series 302 (1961).

[12] H. Kesten and F. Spitzer, A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25. | Zbl 0396.60037

[13] A.J. Lee, $U$-statistics. Theory and practice. Marcel Dekker, Inc., New York (1990). | MR 1075417 | Zbl 0771.62001

[14] M. Maejima, Limit theorems related to a class of operator-self-similar processes. Nagoya Math. J. 142 (1996) 161-181. | Zbl 0865.60033

[15] S. Martínez and D. Petritis, Thermodynamics of a Brownian bridge polymer model in a random environment. J. Phys. A 29 (1996) 1267-1279. | Zbl 0919.60078

[16] I. Meilijson, Mixing properties of a class of skew-products. Israel J. Math. 19 (1974) 266-270. | Zbl 0305.28008

[17] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, Berlin. Fundamental Principles of Mathematical Sciences 293 (1999). | MR 1725357 | Zbl 0917.60006

[18] R.J. Serfling, Approximation theorems of mathematical statistics. John Wiley & Sons Inc., New York. Wiley Series in Probability and Mathematical Statistics (1980). | MR 595165 | Zbl 0538.62002

[19] F. Spitzer, Principles of random walks. Springer-Verlag, New York, second edition. Graduate Texts in Mathematics 34 (1976). | MR 388547 | Zbl 0359.60003

Cité par Sources :