Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T09:23:16.507Z Has data issue: false hasContentIssue false
  • English
  • Français

A concentration inequality for interval maps with an indifferent fixed point

Published online by Cambridge University Press:  01 August 2009

J.-R. CHAZOTTES ,
P. COLLET ,
F. REDIG  and
E. VERBITSKIY
J.-R. CHAZOTTES
Affiliation:
Centre de Physique Théorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France (email: [email protected], [email protected])
P. COLLET
Affiliation:
Centre de Physique Théorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France (email: [email protected], [email protected])
F. REDIG
Affiliation:
Mathematisch Instituut Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands (email: [email protected])
E. VERBITSKIY
Affiliation:
Philips Research, HTC 36 (M/S 2), 5656 AE Eindhoven, The Netherlands (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of n variables, K:[0,1]n→ℝ, which are separately Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We also present applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1097 - 1117
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barbour, A. D., Gerrard, R. M. and Reinert, G.. Iterates of expanding maps. Probab. Theory Related Fields 116(2) (2000), 151–180.CrossRefGoogle Scholar
[2]Brockwell, P.J. and Davis, R.A.. Time Series: Theory and Methods, 2nd edn. Springer, Berlin, 1991.CrossRefGoogle Scholar
[3]Chazottes, J.-R. and Collet, P.. Almost-sure central limit theorems and the Erdös-Rényi law for expanding maps of the interval. Ergod. Th. & Dynam. Sys. 25 (2005), 419–441.CrossRefGoogle Scholar
[4]Chazottes, J.-R., Collet, P., Kuelske, C. and Redig, F.. Concentration inequalities for random fields via coupling. Probab. Theory Related Fields 137 (2007), 201–225.CrossRefGoogle Scholar
[5]Chazottes, J.-R., Collet, P. and Schmitt, B.. Devroye inequality for a class of non-uniformly hyperbolic dynamical systems. Nonlinearity 18 (2005), 2323–2340.CrossRefGoogle Scholar
[6]Chazottes, J.-R., Collet, P. and Schmitt, B.. Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems. Nonlinearity 18 (2005), 2341–2364.CrossRefGoogle Scholar
[7]Chazottes, J.-R. and Gouëzel, S.. On almost-sure versions of classical limit theorems for dynamical systems. Probab. Theory Related Fields 138 (2007), 195–234.CrossRefGoogle Scholar
[8]Collet, P.. Variance and exponential estimates via coupling. Bull. Braz. Math. Soc. 37 (2006), 461–475.CrossRefGoogle Scholar
[9]Collet, P., Martínez, S. and Schmitt, B.. Exponential inequalities for dynamical measures of expanding maps of the interval. Probab. Theory Related Fields 123 (2002), 301–322.CrossRefGoogle Scholar
[10]Dedecker, J. and Prieur, C.. Some unbounded functions of intermittent maps for which the central limit theorem holds. Preprint, 2007. http://arxiv.org/abs/0712.2726. Latin Amer. J. Probab. Math. Stat. to appear.Google Scholar
[11]Devroye, L. and Lugosi, G.. Combinatorial Methods in Density Estimation. Springer, New York, 2000.Google Scholar
[12]Dudley, R. M.. Real Analysis and Probability (Cambridge Studies in Advanced Mathematics, 74). Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original.CrossRefGoogle Scholar
[13]Hu, H.. Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 24(2) (2004), 495–524.CrossRefGoogle Scholar
[14]Ledoux, M.. The Concentration of Measure Phenomenon (Mathematical Surveys and Monographs, 89). American Mathematical Society, Providence, RI, 2001.Google Scholar
[15]Liverani, C.. Central limit theorem for deterministic systems. International Conference on Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362). Longman, Harlow, 1996, pp. 56–75.Google Scholar
[16]Massart, P.. Concentration inequalities and model selection. École d’été de Probab. de Saint-Flour XXXIII —2003 (Lecture Notes in Mathematics, 1896). Springer, Berlin, 2007.Google Scholar
[17]McDiarmid, C.. Concentration. Probabilistic methods for algorithmic discrete mathematics. Algorithms Combin. 16 (1998), 195–248.CrossRefGoogle Scholar
[18]Melbourne, I. and Nicol, M.. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360(12) (2008), 6661–6676.CrossRefGoogle Scholar
[19]Rey-Bellet, L. and Young, L.-S.. Large deviations in nonuniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28(2) (2008), 587–612.CrossRefGoogle Scholar
[20]Talagrand, M.. Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. Inst. Hautes Études Sci. 81 (1995), 73–205; New concentration inequalities in product spaces. Inventiones Math. 126 (1996) 505–563; A new look at independence. Ann. Probab. 24 (1996) 1–34.CrossRefGoogle Scholar
[21]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.CrossRefGoogle Scholar
[22]Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.CrossRefGoogle Scholar