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 -random walk and ( ξ x ) x be a sequence of independent and identically distributed -valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on 2 with values in . We study the weak convergence of the sequence 𝒰 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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  1. [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. Zbl0863.60032
  2. [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). Zbl0944.60003MR1700749
  3. [3] E. Bolthausen, A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989) 108–115. Zbl0679.60028
  4. [4] E. Boylan, Local times for a class of Markoff processes. Illinois J. Math. 8 (1964) 19–39. Zbl0126.33702
  5. [5] E. Buffet and J.V. Pulé, A model of continuous polymers with random charges. J. Math. Phys. 38 (1997) 5143–5152. Zbl0890.60099
  6. [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. Zbl1075.60018
  7. [7] F. den Hollander, Mixing properties for random walk in random scenery. Ann. Probab. 16 (1988) 1788–1802. Zbl0651.60108
  8. [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. Zbl1049.60041
  9. [9] F. den Hollander and J.E. Steif, Mixing properties of the generalized T , T - 1 -process. J. Anal. Math. 72 (1997) 165–202. Zbl0898.60070
  10. [10] R.K. Getoor and H. Kesten, Continuity of local times for Markov processes. Comp. Math. 24 (1972) 277–303. Zbl0293.60069
  11. [11] W. Hoeffding, The strong law of large numbers for U -statistics. Univ. N. Carolina, Institue of Stat. Mimeo series 302 (1961). 
  12. [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. Zbl0396.60037
  13. [13] A.J. Lee, U -statistics. Theory and practice. Marcel Dekker, Inc., New York (1990). Zbl0771.62001MR1075417
  14. [14] M. Maejima, Limit theorems related to a class of operator-self-similar processes. Nagoya Math. J. 142 (1996) 161–181. Zbl0865.60033
  15. [15] S. Martínez and D. Petritis, Thermodynamics of a Brownian bridge polymer model in a random environment. J. Phys. A 29 (1996) 1267–1279. Zbl0919.60078
  16. [16] I. Meilijson, Mixing properties of a class of skew-products. Israel J. Math. 19 (1974) 266–270. Zbl0305.28008
  17. [17] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, Berlin. Fundamental Principles of Mathematical Sciences 293 (1999). Zbl0917.60006MR1725357
  18. [18] R.J. Serfling, Approximation theorems of mathematical statistics. John Wiley & Sons Inc., New York. Wiley Series in Probability and Mathematical Statistics (1980). Zbl0538.62002MR595165
  19. [19] F. Spitzer, Principles of random walks. Springer-Verlag, New York, second edition. Graduate Texts in Mathematics 34 (1976). Zbl0359.60003MR388547

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