Central limit theorem for stationary products of toral automorphisms

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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

Abstract
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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 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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