# American Institute of Mathematical Sciences

September  2013, 33(9): 3981-4002. doi: 10.3934/dcds.2013.33.3981

## The CLT for rotated ergodic sums and related processes

 1 Dept. of Electrical Engineering, Ben-Gurion University, P.O.B 653 Beer-Sheva 84105, Israel 2 IRMAR, UMR CNRS 6625, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex

Received  May 2012 Revised  December 2012 Published  March 2013

Let $( Ω , Α , \mathbb{P} , \tau )$ be an ergodic dynamical system. The rotated ergodic sums of a function $f$ on $\Omega$ for $\theta \in \mathbb{R}$ are $S_n^θ f : = \sum_{k=0}^{n-1} e^{2\pi i k \theta} f \circ \tau^k, n \geq 1$. Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that $(S_n^\theta f)_{n \geq 1}$ satisfies the CLT for a.e. $\theta$ when $(f\circ \tau^n)$ is a regular process.
Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by $K$-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to $\mathbb{Z}^d$-dynamical systems.
Citation: Guy Cohen, Jean-Pierre Conze. The CLT for rotated ergodic sums and related processes. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3981-4002. doi: 10.3934/dcds.2013.33.3981
##### References:
 [1] P. Billingsley, "Convergence of Probability Measures,", Second edition, (1999).  doi: 10.1002/9780470316962.  Google Scholar [2] G. Cohen and J.-P. Conze, Central limit theorem for commutative semigroups of toral endomorphisms,, preprint, (2012).   Google Scholar [3] H. Cramér and H. Wold, Some theorems on distribution functions,, J. Lond. Math. Soc., S1-11 (1936), 1.  doi: 10.1112/jlms/s1-11.4.290.  Google Scholar [4] J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes,, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 1.  doi: 10.1016/S0246-0203(00)00111-4.  Google Scholar [5] G. K. Eagleson, Some simple conditions for limit theorems to be mixing,, Teor. Verojatnost. i Primenen., 21 (1976), 653.   Google Scholar [6] K. Fukuyama and B. Petit, Le théorème limite central pour les suites de R. C. Baker,, Ergodic Theory Dynam. Systems, 21 (2001), 479.  doi: 10.1017/S0143385701001237.  Google Scholar [7] A. M. Garsia, "Topics in Almost Everywhere Convergence,", Lectures in Advanced Mathematics, 4 (1970).   Google Scholar [8] Y. Guivarc'h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov,, Ann. Inst. H. Poincaré Probab. Statist., 24 (1988), 73.   Google Scholar [9] P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application,", Probability and Mathematical Statistics, (1980).   Google Scholar [10] C. Hoffman, A Markov random field which is $K$ but not Bernoulli,, Israel J. Math., 112 (1999), 249.  doi: 10.1007/BF02773484.  Google Scholar [11] B. Kamiński, The theory of invariant partitions for $\mathbbZ^d$-actions,, Bul. Acad. Pol. Sci. Sér. Sci. Math., 29 (1981), 349.   Google Scholar [12] B. Sz.-Nagy and C. Foiaş, "Harmonic Analysis of Operators on Hilbert Space,", Translated from the French and revised North-Holland Publishing Co., (1970).   Google Scholar [13] M. Peligrad and C. Peligrad, On the invariance principle under martingale approximation,, Stoch. Dyn., 11 (2011), 95.  doi: 10.1142/S0219493711003188.  Google Scholar [14] M. Peligrad and W. B. Wu, Central limit theorem for Fourier transforms of stationary processes,, Ann. Probab., 38 (2010), 2009.  doi: 10.1214/10-AOP530.  Google Scholar [15] V. A. Rohlin, Lectures on the entropy of measure-preserving transformations,, Russian Mathematical Surveys, 22 (1967), 1.   Google Scholar [16] K. Schmidt, "Dynamical Systems of Algebraic Origin,", Progress in Mathematics, 128 (1995).   Google Scholar [17] K. Schmidt, On joint recurrence,, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 837.  doi: 10.1016/S0764-4442(99)80115-3.  Google Scholar [18] R. Zweimüller, Mixing limit theorems for ergodic transformations,, J. Theor. Prob., 20 (2007), 1059.  doi: 10.1007/s10959-007-0085-y.  Google Scholar [19] A. Zygmund, "Trigonometric Series: Vols. I, II,", Second edition, (1968).   Google Scholar

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##### References:
 [1] P. Billingsley, "Convergence of Probability Measures,", Second edition, (1999).  doi: 10.1002/9780470316962.  Google Scholar [2] G. Cohen and J.-P. Conze, Central limit theorem for commutative semigroups of toral endomorphisms,, preprint, (2012).   Google Scholar [3] H. Cramér and H. Wold, Some theorems on distribution functions,, J. Lond. Math. Soc., S1-11 (1936), 1.  doi: 10.1112/jlms/s1-11.4.290.  Google Scholar [4] J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes,, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 1.  doi: 10.1016/S0246-0203(00)00111-4.  Google Scholar [5] G. K. Eagleson, Some simple conditions for limit theorems to be mixing,, Teor. Verojatnost. i Primenen., 21 (1976), 653.   Google Scholar [6] K. Fukuyama and B. Petit, Le théorème limite central pour les suites de R. C. Baker,, Ergodic Theory Dynam. Systems, 21 (2001), 479.  doi: 10.1017/S0143385701001237.  Google Scholar [7] A. M. Garsia, "Topics in Almost Everywhere Convergence,", Lectures in Advanced Mathematics, 4 (1970).   Google Scholar [8] Y. Guivarc'h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov,, Ann. Inst. H. Poincaré Probab. Statist., 24 (1988), 73.   Google Scholar [9] P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application,", Probability and Mathematical Statistics, (1980).   Google Scholar [10] C. Hoffman, A Markov random field which is $K$ but not Bernoulli,, Israel J. Math., 112 (1999), 249.  doi: 10.1007/BF02773484.  Google Scholar [11] B. Kamiński, The theory of invariant partitions for $\mathbbZ^d$-actions,, Bul. Acad. Pol. Sci. Sér. Sci. Math., 29 (1981), 349.   Google Scholar [12] B. Sz.-Nagy and C. Foiaş, "Harmonic Analysis of Operators on Hilbert Space,", Translated from the French and revised North-Holland Publishing Co., (1970).   Google Scholar [13] M. Peligrad and C. Peligrad, On the invariance principle under martingale approximation,, Stoch. Dyn., 11 (2011), 95.  doi: 10.1142/S0219493711003188.  Google Scholar [14] M. Peligrad and W. B. Wu, Central limit theorem for Fourier transforms of stationary processes,, Ann. Probab., 38 (2010), 2009.  doi: 10.1214/10-AOP530.  Google Scholar [15] V. A. Rohlin, Lectures on the entropy of measure-preserving transformations,, Russian Mathematical Surveys, 22 (1967), 1.   Google Scholar [16] K. Schmidt, "Dynamical Systems of Algebraic Origin,", Progress in Mathematics, 128 (1995).   Google Scholar [17] K. Schmidt, On joint recurrence,, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 837.  doi: 10.1016/S0764-4442(99)80115-3.  Google Scholar [18] R. Zweimüller, Mixing limit theorems for ergodic transformations,, J. Theor. Prob., 20 (2007), 1059.  doi: 10.1007/s10959-007-0085-y.  Google Scholar [19] A. Zygmund, "Trigonometric Series: Vols. I, II,", Second edition, (1968).   Google Scholar
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