The Annals of Probability

A vector-valued almost sure invariance principle for hyperbolic dynamical systems

Ian Melbourne and Matthew Nicol

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Abstract

We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Hölder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modeled by Young towers with moderate tail decay rates.

In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a two-dimensional Brownian motion.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 478-505.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099919

Digital Object Identifier
doi:10.1214/08-AOP410

Mathematical Reviews number (MathSciNet)
MR2510014

Zentralblatt MATH identifier
1176.37006

Subjects
Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D50: Hyperbolic systems with singularities (billiards, etc.) 60F17: Functional limit theorems; invariance principles

Keywords
Almost sure invariance principle nonuniform hyperbolicity Lorentz gases Brownian motion Young towers

Citation

Melbourne, Ian; Nicol, Matthew. A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 (2009), no. 2, 478--505. doi:10.1214/08-AOP410. https://projecteuclid.org/euclid.aop/1241099919


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References

  • [1] Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs 50. Amer. Math. Soc., Providence, RI.
  • [2] Aaronson, J. and Denker, M. (2001). Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1 193–237.
  • [3] Berger, E. (1990). An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables. Probab. Theory Related Fields 84 161–201.
  • [4] Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29–54.
  • [5] Bunimovich, L. A. and Sinaĭ, Y. G. (1980/81). Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 479–497.
  • [6] Bunimovich, L. A., Sinaĭ, Y. G. and Chernov, N. I. (1991). Statistical properties of two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk 46 43–92, 192.
  • [7] Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19–42.
  • [8] Chernov, N. (2006). Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122 1061–1094.
  • [9] Chernov, N. and Dolgopyat, D. (2006). Hyperbolic billiards and statistical physics. In International Congress of Mathematicians II 1679–1704. Eur. Math. Soc., Zürich.
  • [10] Chernov, N. and Young, L. S. (2000). Decay of correlations for Lorentz gases and hard balls. In Hard Ball Systems and the Lorentz Gas. Encyclopaedia Math. Sci. 101 89–120. Springer, Berlin.
  • [11] Chernov, N. and Zhang, H.-K. (2005). Billiards with polynomial mixing rates. Nonlinearity 18 1527–1553.
  • [12] Chow, Y. S. (1965). Local convergence of martingales and the law of large numbers. Ann. Math. Statist. 36 552–558.
  • [13] Conze, J.-P. and Le Borgne, S. (2001). Méthode de martingales et flot géodésique sur une surface de courbure constante négative. Ergodic Theory Dynam. Systems 21 421–441.
  • [14] Csörgő, M. and Révész, P. (1975). A new method to prove Strassen type laws of invariance principle. II. Z. Wahrsch. Verw. Gebiete 31 261–269.
  • [15] Davydov, J. A. (1970). The invariance principle for stationary processes. Teor. Verojatnost. i Primenen. 15 498–509.
  • [16] 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.
  • [17] Dolgopyat, D. (2004). Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 1637–1689 (electronic).
  • [18] Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables. In Proc. 6th Berkeley Sympos. Math. Statist. Probab. II 513–535. Univ. California Press, Berkeley.
  • [19] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. II. Wiley, New York.
  • [20] 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.
  • [21] Gál, I. S. and Koksma, J. F. (1950). Sur l’ordre de grandeur des fonctions sommables. Nederl. Akad. Wetensch. Proc. 53 638–653. [Indagationes Math. 12 (1950) 192–207.]
  • [22] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • [23] Hofbauer, F. and Keller, G. (1982). Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 119–140.
  • [24] Holland, M. and Melbourne, I. (2007). Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. 76 345–364.
  • [25] Jain, N. C., Jogdeo, K. and Stout, W. F. (1975). Upper and lower functions for martingales and mixing processes. Ann. Probab. 3 119–145.
  • [26] Kiefer, J. (1972). Skorohod embedding of multivariate RV’s, and the sample DF. Z. Wahrsch. Verw. Gebiete 24 1–35.
  • [27] Kuelbs, J. and Philipp, W. (1980). Almost sure invariance principles for partial sums of mixing B-valued random variables. Ann. Probab. 8 1003–1036.
  • [28] Melbourne, I. (2007). Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Amer. Math. Soc. 359 2421–2441 (electronic).
  • [29] Melbourne, I. and Nicol, M. (2005). Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 131–146.
  • [30] 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.
  • [31] Melbourne, I. and Török, A. (2004). Statistical limit theorems for suspension flows. Israel J. Math. 144 191–209.
  • [32] Morales, C. A., Pacifico, M. J. and Pujals, E. R. (1999). Singular hyperbolic systems. Proc. Amer. Math. Soc. 127 3393–3401.
  • [33] Morrow, G. and Philipp, W. (1982). An almost sure invariance principle for Hilbert space valued martingales. Trans. Amer. Math. Soc. 273 231–251.
  • [34] Nagayama, N. (2004). Almost sure invariance principle for dynamical systems with stretched exponential mixing rates. Hiroshima Math. J. 34 371–411.
  • [35] Parry, W. and Pollicott, M. (1990). Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187-188 268.
  • [36] Peligrad, M., Utev, S. and Wu, W. B. (2007). A maximal Lp-inequality for stationary sequences and its applications. Proc. Amer. Math. Soc. 135 541–550 (electronic).
  • [37] Philipp, W. and Stout, W. F. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161. Amer. Math. Soc., Providence, RI.
  • [38] Sinaĭ, J. G. (1970). Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25 141–192.
  • [39] Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 747–817.
  • [40] Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3 211–226.
  • [41] Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) II 315–343. Univ. California Press, Berkeley.
  • [42] Szász, D. and Varjú, T. (2007). Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129 59–80.
  • [43] Volkonskiĭ, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions. I. Theor. Probab. Appl. 4 178–197.
  • [44] Young, L.-S. (1998). Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 585–650.
  • [45] Young, L.-S. (1999). Recurrence times and rates of mixing. Israel J. Math. 110 153–188.