J-R Chazottes et al 2005 Nonlinearity 18 2341 doi:10.1088/0951-7715/18/5/024
Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems
J-R Chazottes1, P Collet1 and B Schmitt2
Show affiliationsRecommended by L Bunimovich
In this paper, we apply the Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of the empirical measure, the shadowing property and the almost-sure central limit theorem. We proved (Chazottes et al 2005 Nonlinearity 18 2323–40) that the Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in Young (1998 Ann. Math. 147 585–650). In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functions, then it holds uniformly in the function spaces. In the last appendix we prove that, for the subclass of one-dimensional systems studied in Young, the density of the absolutely continuous invariant measure belongs to a Besov space.
- PACS
-
05.45.-a Nonlinear dynamics and nonlinear dynamical systems
02.50.-r Probability theory, stochastic processes, and statistics
02.30.Cj Measure and integration
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
- MSC
-
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
- Subjects
- Dates
-
Issue 5 (September 2005)
Received 8 December 2004, in final form 6 June 2005
Published 15 July 2005
-
Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems
J-R Chazottes et al 2005 Nonlinearity 18 2341
-
Wetting of thin liquid films at nanoscale heterogeneous surfaces
K. Mougin and H. Haidara 2003 Europhys. Lett. 61 660
-
Third quantization: a general method to solve master equations for quadratic open Fermi systems
Tomaž Prosen 2008 New J. Phys. 10 043026
-
Key to better qualitative diagnostic calibrations in respiratory inductive plethysmography
R K Millard 2002 Physiol. Meas. 23 N1
-
An ultimate bound on the trajectories of the Lorenz system and its applications
A Yu Pogromsky et al 2003 Nonlinearity 16 1597
-
Valentin Evgen'evich Voskresenskii (on his sixtieth birthday)
Vasilii A Iskovskikh et al 1988 Russ. Math. Surv. 43 201
-
Development of a quantitative assessment system for correlation analysis of footprint parameters to postural control in children
Chi-Hsuan Lin et al 2006 Physiol. Meas. 27 119
-
Abelian duality, walls and boundary conditions in diverse dimensions
Anton Kapustin and Mikhail Tikhonov JHEP11(2009)006
-
A new type of three-finger micro-tweezers
Hao Xing et al 2006 Meas. Sci. Technol. 17 510
-
Structural investigation of MoS2 core–shell nanoparticles formed by an arc discharge in water
I Alexandrou et al 2003 Nanotechnology 14 913
Users also read
What's this?Related review articles
What's this?- Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems—an analytical theory
- Dimension of non-conformal repellers: a survey
- Progress in normal form theory
, and Chern–Simons–Higgs solitons on 
: dimensional reduction of Chern–Pontryagin densities