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 & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597

References:

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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

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