On weak convergence of sampled dynamical systems

Nadine Guillotin-Plantard[1]

  • [1] Université Claude Bernard- Lyon 1, LaPCS - 50 avenue Tony Garnier, 69366 Lyon Cedex 07 (France)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 1, page 211-233
  • ISSN: 0373-0956

Abstract

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Let T α be a rotation on the circle by an irrational angle α , let ( S k ) k ≥ 0 be a transient ℤ -random walk. Let f ∈ L 2 ( μ ) and H ∈ ] 0 , 1 [ , we study the weak convergence of the sequence 1 n H ∑ k = 0 [ n t ] - 1 f ∘ T α S k , n ≥ 1 .

How to cite

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Guillotin-Plantard, Nadine. "Sur la convergence faible des systèmes dynamiques échantillonnés." Annales de l’institut Fourier 54.1 (2004): 211-233. <http://eudml.org/doc/116106>.

@article{Guillotin2004,
abstract = {Soit $T_\{\alpha \}$ la rotation sur le cercle d’angle irrationnel $\alpha $, soit $(S_\{k\})_\{k\ge 0\}$ une marche aléatoire transiente sur $\mathbb \{Z\}$. Soit $f\in L^\{2\}(\mu )$ et $H\in \ ]0,1[$, nous étudions la convergence faible de la suite $\{1\over \{n^\{H\}\}\}\sum _\{k=0\}^\{[nt]-1\}f\circ T_\{\alpha \}^\{S_\{k\}\},\ \ n\ge 1.$},
affiliation = {Université Claude Bernard- Lyon 1, LaPCS - 50 avenue Tony Garnier, 69366 Lyon Cedex 07 (France)},
author = {Guillotin-Plantard, Nadine},
journal = {Annales de l’institut Fourier},
keywords = {dynamical system; random walk; fractional brownian motion; weak convergence},
language = {fre},
number = {1},
pages = {211-233},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sur la convergence faible des systèmes dynamiques échantillonnés},
url = {http://eudml.org/doc/116106},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Guillotin-Plantard, Nadine
TI - Sur la convergence faible des systèmes dynamiques échantillonnés
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 1
SP - 211
EP - 233
AB - Soit $T_{\alpha }$ la rotation sur le cercle d’angle irrationnel $\alpha $, soit $(S_{k})_{k\ge 0}$ une marche aléatoire transiente sur $\mathbb {Z}$. Soit $f\in L^{2}(\mu )$ et $H\in \ ]0,1[$, nous étudions la convergence faible de la suite ${1\over {n^{H}}}\sum _{k=0}^{[nt]-1}f\circ T_{\alpha }^{S_{k}},\ \ n\ge 1.$
LA - fre
KW - dynamical system; random walk; fractional brownian motion; weak convergence
UR - http://eudml.org/doc/116106
ER -

References

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  1. R. Burton, M. Denker, On the central limit theorem for dynamical systems, Trans. of the Amer. Math. Soc. 302 (1987), 715-726 Zbl0628.60030MR891642
  2. N. Guillotin-Plantard, Sur la convergence faible des systèmes dynamiques échantillonnés, C. R. Acad. Sci., Paris 333 (2001), 583-588 Zbl0996.60032MR1860934
  3. J.-P. Kahane, Some random series of functions, (1985), Cambridge University Press Zbl0571.60002MR833073
  4. L. Kuipers, et H. Niederreiter, Uniform distribution of sequences, (1974), Wiley and sons Zbl0281.10001MR419394
  5. M. Lacey, On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, Journal d'Analyse Mathématique 61 (1993), 47-59 Zbl0790.60027MR1253438
  6. M. Lacey, K. Petersen, M. Wierdl, D. Rudolph, Random ergodic theorems with universally representative sequences, Annales de l'institut Henri Poincaré 30 (1994), 353-395 Zbl0813.28004MR1288356
  7. F.-L. Spitzer, Principles of random walks, (1976), Springer, New York Zbl0359.60003MR388547
  8. D. Volný, Invariance principles and Gaussian approximation for strictly stationary processes, Trans. Amer. Math. Soc. 351 (1999), 3351-3371 Zbl0939.37006MR1624218
  9. A. Zygmund, Trigonometric series, (1959), Cambridge University Press Zbl0085.05601MR107776

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