Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Almost sure invariance principle for sequential and non-stationary dynamical systems


Authors: Nicolai Haydn, Matthew Nicol, Andrew Török and Sandro Vaienti
Journal: Trans. Amer. Math. Soc. 369 (2017), 5293-5316
MSC (2010): Primary 37C99
DOI: https://doi.org/10.1090/tran/6812
Published electronically: January 9, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish almost sure invariance principles, a strong form of approximation by Brownian motion, for non-stationary time-series arising as observations on dynamical systems. Our examples include observations on sequential expanding maps, perturbed dynamical systems, non-stationary sequences of functions on hyperbolic systems as well as applications to the shrinking target problem in expanding systems.


References [Enhancements On Off] (What's this?)

  • [1] Roy Adler and Leopold Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 229-334. MR 1085823, https://doi.org/10.1090/S0273-0979-1991-16076-3
  • [2] José F. Alves, Jorge M. Freitas, Stefano Luzzatto, and Sandro Vaienti, From rates of mixing to recurrence times via large deviations, Adv. Math. 228 (2011), no. 2, 1203-1236. MR 2822221, https://doi.org/10.1016/j.aim.2011.06.014
  • [3] Hale Aytaç, Jorge Milhazes Freitas, and Sandro Vaienti, Laws of rare events for deterministic and random dynamical systems, Trans. Amer. Math. Soc. 367 (2015), no. 11, 8229-8278. MR 3391915, https://doi.org/10.1090/S0002-9947-2014-06300-9
  • [4] R. Aimino, J. Rousseau, Concentration inequalities for sequential dynamical systems of the unit interval, to appear on Ergod. Th. & Dynam. Sys..
  • [5] Wael Bahsoun, Christopher Bose, and Yuejiao Duan, Decay of correlation for random intermittent maps, Nonlinearity 27 (2014), no. 7, 1543-1554. MR 3225871, https://doi.org/10.1088/0951-7715/27/7/1543
  • [6] Wael Bahsoun and Sandro Vaienti, Escape rates formulae and metastability for randomly perturbed maps, Nonlinearity 26 (2013), no. 5, 1415-1438. MR 3056132, https://doi.org/10.1088/0951-7715/26/5/1415
  • [7] Daniel Berend and Vitaly Bergelson, Ergodic and mixing sequences of transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 353-366. MR 776873, https://doi.org/10.1017/S0143385700002509
  • [8] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
  • [9] Romain Aimino, Matthew Nicol, and Sandro Vaienti, Annealed and quenched limit theorems for random expanding dynamical systems, Probab. Theory Related Fields 162 (2015), no. 1-2, 233-274. MR 3350045, https://doi.org/10.1007/s00440-014-0571-y
  • [10] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001), 1-27. MR 1826488, https://doi.org/10.1007/BF02809888
  • [11] P. Collet, An estimate of the decay of correlations for mixing non Markov expanding maps of the interval, preprint (1984).
  • [12] Jean-Pierre Conze and Albert Raugi, Limit theorems for sequential expanding dynamical systems on $ [0,1]$, Ergodic theory and related fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 89-121. MR 2331327, https://doi.org/10.1090/conm/430/08253
  • [13] Christophe Cuny and Florence Merlevède, Strong invariance principles with rate for ``reverse'' martingale differences and applications, J. Theoret. Probab. 28 (2015), no. 1, 137-183. MR 3320963, https://doi.org/10.1007/s10959-013-0506-z
  • [14] Michael Field, Ian Melbourne, and Andrew Török, Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 87-110. MR 1971198, https://doi.org/10.1017/S0143385702000901
  • [15] M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739-741 (Russian). MR 0251785
  • [16] Sébastien Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields 128 (2004), no. 1, 82-122. MR 2027296, https://doi.org/10.1007/s00440-003-0300-4
  • [17] Sébastien Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Probab. 38 (2010), no. 4, 1639-1671. MR 2663640, https://doi.org/10.1214/10-AOP525
  • [18] Cecilia González-Tokman, Brian R. Hunt, and Paul Wright, Approximating invariant densities of metastable systems, Ergodic Theory Dynam. Systems 31 (2011), no. 5, 1345-1361. MR 2832249, https://doi.org/10.1017/S0143385710000337
  • [19] Chinmaya Gupta, William Ott, and Andrei Török, Memory loss for time-dependent piecewise expanding systems in higher dimension, Math. Res. Lett. 20 (2013), no. 1, 141-161. MR 3126728, https://doi.org/10.4310/MRL.2013.v20.n1.a12
  • [20] Huyi Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 495-524. MR 2054191, https://doi.org/10.1017/S0143385703000671
  • [21] Huyi Hu and Sandro Vaienti, Absolutely continuous invariant measures for non-uniformly expanding maps, Ergodic Theory Dynam. Systems 29 (2009), no. 4, 1185-1215. MR 2529645, https://doi.org/10.1017/S0143385708000576
  • [22] Nicolai Haydn, Matthew Nicol, Sandro Vaienti, and Licheng Zhang, Central limit theorems for the shrinking target problem, J. Stat. Phys. 153 (2013), no. 5, 864-887. MR 3124980, https://doi.org/10.1007/s10955-013-0860-3
  • [23] Carlangelo Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys. 78 (1995), no. 3-4, 1111-1129. MR 1315241, https://doi.org/10.1007/BF02183704
  • [24] Carlangelo Liverani, Benoît Saussol, and Sandro Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671-685. MR 1695915, https://doi.org/10.1017/S0143385799133856
  • [25] Ian Melbourne and Matthew Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Comm. Math. Phys. 260 (2005), no. 1, 131-146. MR 2175992, https://doi.org/10.1007/s00220-005-1407-5
  • [26] Ian Melbourne and Matthew Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems, Ann. Probab. 37 (2009), no. 2, 478-505. MR 2510014, https://doi.org/10.1214/08-AOP410
  • [27] Florence Merlevède and Emmanuel Rio, Strong approximation of partial sums under dependence conditions with application to dynamical systems, Stochastic Process. Appl. 122 (2012), no. 1, 386-417. MR 2860454, https://doi.org/10.1016/j.spa.2011.08.012
  • [28] Péter Nándori, Domokos Szász, and Tamás Varjú, A central limit theorem for time-dependent dynamical systems, J. Stat. Phys. 146 (2012), no. 6, 1213-1220. MR 2903045, https://doi.org/10.1007/s10955-012-0451-8
  • [29] William Ott, Mikko Stenlund, and Lai-Sang Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett. 16 (2009), no. 3, 463-475. MR 2511626, https://doi.org/10.4310/MRL.2009.v16.n3.a7
  • [30] William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
  • [31] Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 issue 2, 161 (1975), iv+140. MR 0433597
  • [32] Omri Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), no. 3, 629-653. MR 1946554, https://doi.org/10.1007/s00222-002-0248-5
  • [33] Weixiao Shen and Sebastian van Strien, On stochastic stability of expanding circle maps with neutral fixed points, Dyn. Syst. 28 (2013), no. 3, 423-452. MR 3170624, https://doi.org/10.1080/14689367.2013.806733
  • [34] Vladimir G. Sprindžuk, Metric theory of Diophantine approximations, V. H. Winston & Sons, Washington, D.C.; A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. Translated from the Russian and edited by Richard A. Silverman; With a foreword by Donald J. Newman; Scripta Series in Mathematics. MR 548467
  • [35] Mikko Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity 24 (2011), no. 10, 2991-3018. MR 2842105, https://doi.org/10.1088/0951-7715/24/10/016
  • [36] Mikko Stenlund, Lai-Sang Young, and Hongkun Zhang, Dispersing billiards with moving scatterers, Comm. Math. Phys. 322 (2013), no. 3, 909-955. MR 3079336, https://doi.org/10.1007/s00220-013-1746-6
  • [37] M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium 1997, IMPA.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37C99

Retrieve articles in all journals with MSC (2010): 37C99


Additional Information

Nicolai Haydn
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: nhaysdn@math.usc.edu

Matthew Nicol
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204
Email: nicol@math.uh.edu

Andrew Török
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204 – and – Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania
Email: torok@math.uh.edu

Sandro Vaienti
Affiliation: Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France
Email: vaienti@cpt.univ-mrs.fr

DOI: https://doi.org/10.1090/tran/6812
Received by editor(s): June 17, 2014
Received by editor(s) in revised form: August 18, 2015, and August 20, 2015
Published electronically: January 9, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society