Vector-valued almost sure invariance principles for (non)stationary and random dynamical systems

Su, Yaofeng

doi:10.1090/tran/8609

Vector-valued almost sure invariance principles for (non)stationary and random dynamical systems
HTML articles powered by AMS MathViewer

by Yaofeng Su PDF

Trans. Amer. Math. Soc. 375 (2022), 4809-4848 Request permission

Abstract:

We study the limit behavior of (non)stationary and random chaotic dynamical systems. Several (vector-valued) almost sure invariance principles for (non)stationary dynamical systems and quenched (vector-valued) almost sure invariance principles for random dynamical systems are proved. We also apply our results to stationary chaotic dynamical systems, which admit Young towers, and to (non)uniformly expanding non-stationary and random dynamical systems with intermittencies or uniform spectral gaps. It implies that the systems under study tend to a Brownian motion under various scalings.

References

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, and Sandro Vaienti, Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 793–806. MR 3277171, DOI 10.3934/dcds.2015.35.793
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
István Berkes and Walter Philipp, Approximation theorems for independent and weakly dependent random vectors, Ann. Probab. 7 (1979), no. 1, 29–54. MR 515811
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
C. Cuny, J. Dedecker, A. Korepanov, and F. Merlevède, Rates in almost sure invariance principle for quickly mixing dynamical systems, Stoch. Dyn. 20 (2020), no. 1, 2050002, 28. MR 4066797, DOI 10.1142/S0219493720500021
C. Cuny, J. Dedecker, A. Korepanov, and F. Merlevède, Rates in almost sure invariance principle for slowly mixing dynamical systems, Ergodic Theory Dynam. Systems 40 (2020), no. 9, 2317–2348. MR 4130806, DOI 10.1017/etds.2019.2
D. Dragičević, G. Froyland, C. González-Tokman, and S. Vaienti, Almost sure invariance principle for random piecewise expanding maps, Nonlinearity 31 (2018), no. 5, 2252–2280. MR 3816673, DOI 10.1088/1361-6544/aaaf4b
D. Dragičević, G. Froyland, C. González-Tokman, and S. Vaienti, A spectral approach for quenched limit theorems for random expanding dynamical systems, Comm. Math. Phys. 360 (2018), no. 3, 1121–1187. MR 3803820, DOI 10.1007/s00220-017-3083-7
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
Nicolai Haydn, Matthew Nicol, Andrew Török, and Sandro Vaienti, Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5293–5316. MR 3646763, DOI 10.1090/tran/6812
Olli Hella and Juho Leppänen, Central limit theorems with a rate of convergence for time-dependent intermittent maps, Stoch. Dyn. 20 (2020), no. 4, 2050025, 28. MR 4128743, DOI 10.1142/S0219493720500252
Olli Hella and Mikko Stenlund, Quenched normal approximation for random sequences of transformations, J. Stat. Phys. 178 (2020), no. 1, 1–37. MR 4056650, DOI 10.1007/s10955-019-02390-5
Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169, DOI 10.1007/978-1-4757-4015-8
Alexey Korepanov, Rates in almost sure invariance principle for dynamical systems with some hyperbolicity, Comm. Math. Phys. 363 (2018), no. 1, 173–190. MR 3849987, DOI 10.1007/s00220-018-3234-5
J. Kuelbs and Walter Philipp, Almost sure invariance principles for partial sums of mixing $B$-valued random variables, Ann. Probab. 8 (1980), no. 6, 1003–1036. MR 602377, DOI 10.1214/aop/1176994565
Carlangelo Liverani, Central limit theorem for deterministic systems, International Conference on Dynamical Systems (Montevideo, 1995) Pitman Res. Notes Math. Ser., vol. 362, Longman, Harlow, 1996, pp. 56–75. MR 1460797
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
Stefano Luzzatto and Ian Melbourne, Statistical properties and decay of correlations for interval maps with critical points and singularities, Comm. Math. Phys. 320 (2013), no. 1, 21–35. MR 3046988, DOI 10.1007/s00220-013-1709-y
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
Matthew Nicol, Felipe Perez Pereira, and Andrew Török, Large deviations and central limit theorems for sequential and random systems of intermittent maps, Ergodic Theory Dynam. Systems 41 (2021), no. 9, 2805–2832. MR 4297074, DOI 10.1017/etds.2020.90
Matthew Nicol, Andrew Török, and Sandro Vaienti, Central limit theorems for sequential and random intermittent dynamical systems, Ergodic Theory Dynam. Systems 38 (2018), no. 3, 1127–1153. MR 3784257, DOI 10.1017/etds.2016.69
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
Marek Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), no. 1, 69–80. MR 728198, DOI 10.4064/sm-76-1-69-80
R. J. Serfling, Contributions to central limit theory for dependent variables, Ann. Math. Statist. 39 (1968), 1158–1175. MR 228053, DOI 10.1214/aoms/1177698240
R. J. Serfling, Moment inequalities for the maximum cumulative sum, Ann. Math. Statist. 41 (1970), 1227–1234. MR 268938, DOI 10.1214/aoms/1177696898
Yaofeng Su, Almost surely invariance principle for non-stationary and random intermittent dynamical systems, Discrete Contin. Dyn. Syst. 39 (2019), no. 11, 6585–6597. MR 4026993, DOI 10.3934/dcds.2019286
Yaofeng Su, Random young towers and quenched limit laws, e-prints, arXiv:1907.12199, 2019.
Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180