# American Institute of Mathematical Sciences

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, 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

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##### 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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