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

Central limit theorem for stationary products of toral automorphisms

Jean-Pierre Conze 1, , Stéphane Le Borgne 1, and  Mikaël Roger 1,

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.
Keywords: kicked stationary process, multiplicative system, Lyapunov exponents., Toral automorphisms, central limit theorem.
Mathematics Subject Classification: 60F05, 37A30, 60G10, 37D9.
Citation: Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems, 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), 267-289.  Google Scholar

[2]

A. Ayyer and M. Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms, Chaos, 17 (2007), 043116, 7 pp.  Google Scholar

[3]

A. Ayyer, C. Liverani and M. Stenlund, Quenched CLT for random toral automorphism, Discrete Contin. Dyn. Syst., 24 (2009), 331-348. 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), 247-279 and 617-627, (MR1286845).  Google Scholar

[5]

I. Berkes, On the asymptotic behaviour of $Sf(n_kx)$. Main theorems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34 (1976), 319-345. doi: 10.1007/BF00535967.  Google Scholar

[6]

B. M. Brown, Martingale central limit theorems, Annals of Math. Statistics, 42 (1971), 59-66.  Google Scholar

[7]

J.-P. Conze and S. Le Borgne, Limit law for some modified ergodic sums, Stochastics and Dynamics, 11 (2011), 107-133. 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 "Ergodic Theory and Related Fields," Contemporary Mathematics, 430, Amer. Math. Soc., Providence, RI, (2007), 89-121.  Google Scholar

[10]

W. Feller, "An Introduction to Probability Theory and its Applications," Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.  Google Scholar

[11]

V. F. Gaposhkin, Lacunary series and independent functions, (Russian), Uspehi Mat. Nauk, 21 (1966), 3-82. Google Scholar

[12]

J. Komlós, A central limit theorem for multiplicative systems, Canad. Math. Bull., 16 (1973), 67-73. 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), 179-210.  Google Scholar

[14]

B. Petit, Le théorème limite central pour des sommes de Riesz-Raĭkov, Probab. Theory Related Fields, 93 (1992), 407-438. 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), 609-619. doi: 10.1017/S0143385703000531.  Google Scholar

[16]

A. Raugi, "Théorème Ergodique Multiplicatif, Produits de Matrices Aléatoires Indépendantes," Publ. Inst. Rech. Math. Rennes, 1996/1997, Univ. Rennes I, Rennes, 1997. Google Scholar

[17]

M. Viana, "Stochastic Dynamics of Deterministic Systems," Brazilian Math. Colloquium, IMPA, 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), 267-289.  Google Scholar

[2]

A. Ayyer and M. Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms, Chaos, 17 (2007), 043116, 7 pp.  Google Scholar

[3]

A. Ayyer, C. Liverani and M. Stenlund, Quenched CLT for random toral automorphism, Discrete Contin. Dyn. Syst., 24 (2009), 331-348. 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), 247-279 and 617-627, (MR1286845).  Google Scholar

[5]

I. Berkes, On the asymptotic behaviour of $Sf(n_kx)$. Main theorems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34 (1976), 319-345. doi: 10.1007/BF00535967.  Google Scholar

[6]

B. M. Brown, Martingale central limit theorems, Annals of Math. Statistics, 42 (1971), 59-66.  Google Scholar

[7]

J.-P. Conze and S. Le Borgne, Limit law for some modified ergodic sums, Stochastics and Dynamics, 11 (2011), 107-133. 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 "Ergodic Theory and Related Fields," Contemporary Mathematics, 430, Amer. Math. Soc., Providence, RI, (2007), 89-121.  Google Scholar

[10]

W. Feller, "An Introduction to Probability Theory and its Applications," Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.  Google Scholar

[11]

V. F. Gaposhkin, Lacunary series and independent functions, (Russian), Uspehi Mat. Nauk, 21 (1966), 3-82. Google Scholar

[12]

J. Komlós, A central limit theorem for multiplicative systems, Canad. Math. Bull., 16 (1973), 67-73. 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), 179-210.  Google Scholar

[14]

B. Petit, Le théorème limite central pour des sommes de Riesz-Raĭkov, Probab. Theory Related Fields, 93 (1992), 407-438. 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), 609-619. doi: 10.1017/S0143385703000531.  Google Scholar

[16]

A. Raugi, "Théorème Ergodique Multiplicatif, Produits de Matrices Aléatoires Indépendantes," Publ. Inst. Rech. Math. Rennes, 1996/1997, Univ. Rennes I, Rennes, 1997. Google Scholar

[17]

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

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