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 and 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.

[2]

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

[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.

[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).

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[12]

J. Komlós, A central limit theorem for multiplicative systems, Canad. Math. Bull., 16 (1973), 67-73. doi: 10.4153/CMB-1973-014-3.

[13]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[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.

[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.

[16]

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

[17]

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

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.

[2]

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

[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.

[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).

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[12]

J. Komlós, A central limit theorem for multiplicative systems, Canad. Math. Bull., 16 (1973), 67-73. doi: 10.4153/CMB-1973-014-3.

[13]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[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.

[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.

[16]

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

[17]

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

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