August  2005, 5(3): 565-586. doi: 10.3934/dcdsb.2005.5.565

Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources


CNRS-CPHT, École Polytechnique, 91128 Palaiseau Cedex, France


Instituto de Física--UASLP, 78000 San Luis Potosí, SLP, Mexico

Received  January 2004 Revised  November 2004 Published  May 2005

Motivated by entropy estimation from chaotic time series, we pro- vide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a n-cylinder fluctuates in the same way as the inverse measure of this n-cylinder at 'small scales', but in a different way at 'large scales'. In particular, the Rényi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more di±cult to handle theoretically. The results we obtain for return times, though less precise than for hitting times, complete the available ones.
Citation: Jean René Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 565-586. doi: 10.3934/dcdsb.2005.5.565

Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2


Milton Ko. Rényi entropy and recurrence. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403


Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597


James Nolen. A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167


Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477


Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199


Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857


Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143


Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729


Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795


David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873


Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113


Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881


JÓzsef Balogh, Hoi Nguyen. A general law of large permanent. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5285-5297. doi: 10.3934/dcds.2017229


Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012


Emmanuel Schenck. Exponential gaps in the length spectrum. Journal of Modern Dynamics, 2020, 16: 207-223. doi: 10.3934/jmd.2020007


Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103


Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805


Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021021


Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15

2020 Impact Factor: 1.327


  • PDF downloads (47)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]