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]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167

[2]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233

[3]

Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332

[4]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477

[5]

Federico Rodriguez Hertz. Global rigidity of certain Abelian actions by toral automorphisms. Journal of Modern Dynamics, 2007, 1 (3) : 425-442. doi: 10.3934/jmd.2007.1.425

[6]

Lennard F. Bakker, Pedro Martins Rodrigues. A profinite group invariant for hyperbolic toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1965-1976. doi: 10.3934/dcds.2012.32.1965

[7]

Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143

[8]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[9]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957

[10]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

[11]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433

[12]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012

[13]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287

[14]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004

[15]

Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63

[16]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

[17]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861

[18]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619

[19]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149

[20]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

[Back to Top]