The Annals of Probability

Almost sure invariance principle for dynamical systems by spectral methods

Sébastien Gouëzel

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We prove the almost sure invariance principle for stationary ℝd-valued random processes (with very precise dimension-independent error terms), solely under a strong assumption concerning the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains using strong or weak spectral perturbation arguments.

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Ann. Probab., Volume 38, Number 4 (2010), 1639-1671.

First available in Project Euclid: 8 July 2010

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Primary: 60F17: Functional limit theorems; invariance principles 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems

Almost sure invariance principle coupling transfer operator


Gouëzel, Sébastien. Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38 (2010), no. 4, 1639--1671. doi:10.1214/10-AOP525.

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  • [1] Aaronson, J. and Weiss, B. (2000). Remarks on the tightness of cocycles. Colloq. Math. 84/85 363–376.
  • [2] Baladi, V. and Tsujii, M. (2007). Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier (Grenoble) 57 127–154.
  • [3] Bardet, J.-B., Gouëzel, S. and Keller, G. (2007). Limit theorems for coupled interval maps. Stoch. Dyn. 7 17–36.
  • [4] Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29–54.
  • [5] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [6] Denker, M. and Philipp, W. (1984). Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dynam. Systems 4 541–552.
  • [7] Einmahl, U. (1989). Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 20–68.
  • [8] Field, M., Melbourne, I. and Török, A. (2003). Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions. Ergodic Theory Dynam. Systems 23 87–110.
  • [9] Gouëzel, S. (2008). Characterization of weak convergence of Birkhoff sums for Gibbs–Markov maps. Preprint.
  • [10] Gouëzel, S. and Liverani, C. (2008). Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties. J. Differential Geom. 79 433–477.
  • [11] Guivarc’h, Y. and Hardy, J. (1988). Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré Probab. Statist. 24 73–98.
  • [12] Hervé, L. and Pène, F. (2008). Nagaev method via Keller–Liverani theorem. Preprint.
  • [13] Hofbauer, F. and Keller, G. (1982). Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 119–140.
  • [14] Keller, G. and Liverani, C. (1999). Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 141–152.
  • [15] Liverani, C. (2003). Invariant measures and their properties. A functional analytic point of view. In Dynamical Systems. Part II. Pubbl. Cent. Ric. Mat. Ennio Giorgi 185–237. Scuola Norm. Sup., Pisa.
  • [16] Melbourne, I. and Nicol, M. (2009). A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 478–505.
  • [17] Melbourne, I. and Török, A. (2002). Central limit theorems and invariance principles for time-one maps of hyperbolic flows. Comm. Math. Phys. 229 57–71.
  • [18] Melbourne, I. and Török, A. (2004). Statistical limit theorems for suspension flows. Israel J. Math. 144 191–209.
  • [19] Parry, W. and Pollicott, M. (1990). Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187-188 268.
  • [20] Philipp, W. and Stout, W. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2 161 iv+140.
  • [21] Rosenthal, H. P. (1970). On the subspaces of Lp (p > 2) spanned by sequences of independent random variables. Israel J. Math. 8 273–303.
  • [22] Saussol, B. (2000). Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 223–248.
  • [23] Serfling, R. J. (1970). Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41 1227–1234.
  • [24] Tsujii, M. (2008). Quasi-compactness of transfer operators for contact Anosov flows. Preprint.
  • [25] Zaĭtsev, A. Y. (1998). Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. ESAIM Probab. Stat. 2 41–108.
  • [26] Zaĭtsev, A. Y. (2007). Estimates for the rate of strong approximation in the multidimensional invariance principle. J. Math. Sci. (N. Y.) 145 4856–4865.