Almost sure invariance principle for sequential and non-stationary dynamical systems
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- by Nicolai Haydn, Matthew Nicol, Andrew Török and Sandro Vaienti PDF
- Trans. Amer. Math. Soc. 369 (2017), 5293-5316 Request permission
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
- 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, DOI 10.1090/S0273-0979-1991-16076-3
- 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, DOI 10.1016/j.aim.2011.06.014
- 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, DOI 10.1090/S0002-9947-2014-06300-9
- R. Aimino, J. Rousseau, Concentration inequalities for sequential dynamical systems of the unit interval, to appear on Ergod. Th. & Dynam. Sys..
- Wael Bahsoun, Christopher Bose, and Yuejiao Duan, Decay of correlation for random intermittent maps, Nonlinearity 27 (2014), no. 7, 1543–1554. MR 3225871, DOI 10.1088/0951-7715/27/7/1543
- Wael Bahsoun and Sandro Vaienti, Escape rates formulae and metastability for randomly perturbed maps, Nonlinearity 26 (2013), no. 5, 1415–1438. MR 3056132, DOI 10.1088/0951-7715/26/5/1415
- Daniel Berend and Vitaly Bergelson, Ergodic and mixing sequences of transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 353–366. MR 776873, DOI 10.1017/S0143385700002509
- 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
- 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, DOI 10.1007/s00440-014-0571-y
- N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001), 1–27. MR 1826488, DOI 10.1007/BF02809888
- P. Collet, An estimate of the decay of correlations for mixing non Markov expanding maps of the interval, preprint (1984).
- 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, DOI 10.1090/conm/430/08253
- 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, DOI 10.1007/s10959-013-0506-z
- 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, DOI 10.1017/S0143385702000901
- M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
- Sébastien Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields 128 (2004), no. 1, 82–122. MR 2027296, DOI 10.1007/s00440-003-0300-4
- Sébastien Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Probab. 38 (2010), no. 4, 1639–1671. MR 2663640, DOI 10.1214/10-AOP525
- 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, DOI 10.1017/S0143385710000337
- 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, DOI 10.4310/MRL.2013.v20.n1.a12
- 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, DOI 10.1017/S0143385703000671
- 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, DOI 10.1017/S0143385708000576
- 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, DOI 10.1007/s10955-013-0860-3
- Carlangelo Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys. 78 (1995), no. 3-4, 1111–1129. MR 1315241, DOI 10.1007/BF02183704
- Carlangelo Liverani, Benoît Saussol, and Sandro Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671–685. MR 1695915, DOI 10.1017/S0143385799133856
- Ian Melbourne and Matthew Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Comm. Math. Phys. 260 (2005), no. 1, 131–146. MR 2175992, DOI 10.1007/s00220-005-1407-5
- 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, DOI 10.1214/08-AOP410
- 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, DOI 10.1016/j.spa.2011.08.012
- 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, DOI 10.1007/s10955-012-0451-8
- 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, DOI 10.4310/MRL.2009.v16.n3.a7
- 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
- Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 (1975), no. 161,, 161, iv+140. MR 433597, DOI 10.1090/memo/0161
- Omri Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), no. 3, 629–653. MR 1946554, DOI 10.1007/s00222-002-0248-5
- 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, DOI 10.1080/14689367.2013.806733
- Vladimir G. Sprindžuk, Metric theory of Diophantine approximations, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; 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. MR 548467
- Mikko Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity 24 (2011), no. 10, 2991–3018. MR 2842105, DOI 10.1088/0951-7715/24/10/016
- Mikko Stenlund, Lai-Sang Young, and Hongkun Zhang, Dispersing billiards with moving scatterers, Comm. Math. Phys. 322 (2013), no. 3, 909–955. MR 3079336, DOI 10.1007/s00220-013-1746-6
- M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium 1997, IMPA.
Additional Information
- Nicolai Haydn
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
- MR Author ID: 241411
- Email: nhaysdn@math.usc.edu
- Matthew Nicol
- Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204
- MR Author ID: 350236
- 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
- MR Author ID: 249702
- Email: torok@math.uh.edu
- Sandro Vaienti
- Affiliation: Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France
- MR Author ID: 176525
- Email: vaienti@cpt.univ-mrs.fr
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5293-5316
- MSC (2010): Primary 37C99
- DOI: https://doi.org/10.1090/tran/6812
- MathSciNet review: 3646763