May  2012, 32(5): 1597-1626. doi: 10.3934/dcds.2012.32.1597

Central limit theorem for stationary products of toral automorphisms

1. 

IRMAR, UMR CNRS 6625, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France, France, France

Received  June 2010 Revised  October 2011 Published  January 2012

Let $(A_n(\omega))$ be a stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Hölder function $f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega) \, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm \ mod \ } 1)$ and prove a Central Limit Theorem for a.e. $\omega$ in different situations in particular for "kicked" stationary processes. We use the method of multiplicative systems of Komlòs and the Multiplicative Ergodic Theorem.
Citation: Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
References:
[1]

C. Aistleitner and I. Berkes, On the central limit theorem for $f(n_k x)$,, Probab. Theory Related Fields, 146 (2010), 267.   Google Scholar

[2]

A. Ayyer and M. Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms,, Chaos, 17 (2007).   Google Scholar

[3]

A. Ayyer, C. Liverani and M. Stenlund, Quenched CLT for random toral automorphism,, Discrete Contin. Dyn. Syst., 24 (2009), 331.  doi: 10.3934/dcds.2009.24.331.  Google Scholar

[4]

V. I. Bakhtin, Random processes generated by a hyperbolic sequence of mappings (I, II),, Rus. Ac. Sci. Izv. Math., 44 (1995), 247.   Google Scholar

[5]

I. Berkes, On the asymptotic behaviour of $Sf(n_kx)$. Main theorems,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34 (1976), 319.  doi: 10.1007/BF00535967.  Google Scholar

[6]

B. M. Brown, Martingale central limit theorems,, Annals of Math. Statistics, 42 (1971), 59.   Google Scholar

[7]

J.-P. Conze and S. Le Borgne, Limit law for some modified ergodic sums,, Stochastics and Dynamics, 11 (2011), 107.  doi: 10.1142/S021949371100319X.  Google Scholar

[8]

J.-P. Conze and S. Le Borgne, Théorème limite central presque sûr pour les marches aléatoires avec trou spectral, [Quenched central limit theorem for random walks with a spectral gap],, CRAS, (2011).   Google Scholar

[9]

J.-P. Conze and A. Raugi, Limit theorems for sequential expanding dynamical systems on [0,1],, in, 430 (2007), 89.   Google Scholar

[10]

W. Feller, "An Introduction to Probability Theory and its Applications,", Vol. II, (1971).   Google Scholar

[11]

V. F. Gaposhkin, Lacunary series and independent functions,, (Russian), 21 (1966), 3.   Google Scholar

[12]

J. Komlós, A central limit theorem for multiplicative systems,, Canad. Math. Bull., 16 (1973), 67.  doi: 10.4153/CMB-1973-014-3.  Google Scholar

[13]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 179.   Google Scholar

[14]

B. Petit, Le théorème limite central pour des sommes de Riesz-Raĭkov,, Probab. Theory Related Fields, 93 (1992), 407.  doi: 10.1007/BF01192715.  Google Scholar

[15]

L. Polterovich and Z. Rudnick, Stable mixing for cat maps and quasi-morphisms of the modular group,, Ergodic Theory Dynam. Systems, 24 (2004), 609.  doi: 10.1017/S0143385703000531.  Google Scholar

[16]

A. Raugi, "Théorème Ergodique Multiplicatif, Produits de Matrices Aléatoires Indépendantes,", Publ. Inst. Rech. Math. Rennes, (1996).   Google Scholar

[17]

M. Viana, "Stochastic Dynamics of Deterministic Systems,", Brazilian Math. Colloquium, (1997).   Google Scholar

show all references

References:
[1]

C. Aistleitner and I. Berkes, On the central limit theorem for $f(n_k x)$,, Probab. Theory Related Fields, 146 (2010), 267.   Google Scholar

[2]

A. Ayyer and M. Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms,, Chaos, 17 (2007).   Google Scholar

[3]

A. Ayyer, C. Liverani and M. Stenlund, Quenched CLT for random toral automorphism,, Discrete Contin. Dyn. Syst., 24 (2009), 331.  doi: 10.3934/dcds.2009.24.331.  Google Scholar

[4]

V. I. Bakhtin, Random processes generated by a hyperbolic sequence of mappings (I, II),, Rus. Ac. Sci. Izv. Math., 44 (1995), 247.   Google Scholar

[5]

I. Berkes, On the asymptotic behaviour of $Sf(n_kx)$. Main theorems,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34 (1976), 319.  doi: 10.1007/BF00535967.  Google Scholar

[6]

B. M. Brown, Martingale central limit theorems,, Annals of Math. Statistics, 42 (1971), 59.   Google Scholar

[7]

J.-P. Conze and S. Le Borgne, Limit law for some modified ergodic sums,, Stochastics and Dynamics, 11 (2011), 107.  doi: 10.1142/S021949371100319X.  Google Scholar

[8]

J.-P. Conze and S. Le Borgne, Théorème limite central presque sûr pour les marches aléatoires avec trou spectral, [Quenched central limit theorem for random walks with a spectral gap],, CRAS, (2011).   Google Scholar

[9]

J.-P. Conze and A. Raugi, Limit theorems for sequential expanding dynamical systems on [0,1],, in, 430 (2007), 89.   Google Scholar

[10]

W. Feller, "An Introduction to Probability Theory and its Applications,", Vol. II, (1971).   Google Scholar

[11]

V. F. Gaposhkin, Lacunary series and independent functions,, (Russian), 21 (1966), 3.   Google Scholar

[12]

J. Komlós, A central limit theorem for multiplicative systems,, Canad. Math. Bull., 16 (1973), 67.  doi: 10.4153/CMB-1973-014-3.  Google Scholar

[13]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 179.   Google Scholar

[14]

B. Petit, Le théorème limite central pour des sommes de Riesz-Raĭkov,, Probab. Theory Related Fields, 93 (1992), 407.  doi: 10.1007/BF01192715.  Google Scholar

[15]

L. Polterovich and Z. Rudnick, Stable mixing for cat maps and quasi-morphisms of the modular group,, Ergodic Theory Dynam. Systems, 24 (2004), 609.  doi: 10.1017/S0143385703000531.  Google Scholar

[16]

A. Raugi, "Théorème Ergodique Multiplicatif, Produits de Matrices Aléatoires Indépendantes,", Publ. Inst. Rech. Math. Rennes, (1996).   Google Scholar

[17]

M. Viana, "Stochastic Dynamics of Deterministic Systems,", Brazilian Math. Colloquium, (1997).   Google Scholar

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[2]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[3]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[4]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[5]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

[6]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[7]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[8]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[9]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[10]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[11]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[12]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[13]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[14]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[15]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[16]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[17]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[19]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[20]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (1)

[Back to Top]