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, 2013, 33 (9) : 3981-4002. doi: 10.3934/dcds.2013.33.3981
References:
[1]

P. Billingsley, "Convergence of Probability Measures," Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 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) 290-294. 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), 1-34. 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), 653-660.  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), 479-492. doi: 10.1017/S0143385701001237.  Google Scholar

[7]

A. M. Garsia, "Topics in Almost Everywhere Convergence," Lectures in Advanced Mathematics, 4, Markham Publishing Co., Chicago, Ill., 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), 73-98.  Google Scholar

[9]

P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application," Probability and Mathematical Statistics, Academic Press, Inc., New York-London, 1980.  Google Scholar

[10]

C. Hoffman, A Markov random field which is $K$ but not Bernoulli, Israel J. Math., 112 (1999), 249-269. 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), 349-362.  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., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970.  Google Scholar

[13]

M. Peligrad and C. Peligrad, On the invariance principle under martingale approximation, Stoch. Dyn., 11 (2011), 95-105. 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), 2009-2022. doi: 10.1214/10-AOP530.  Google Scholar

[15]

V. A. Rohlin, Lectures on the entropy of measure-preserving transformations, Russian Mathematical Surveys, 22 (1967), 1-52. Google Scholar

[16]

K. Schmidt, "Dynamical Systems of Algebraic Origin," Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[17]

K. Schmidt, On joint recurrence, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 837-842. doi: 10.1016/S0764-4442(99)80115-3.  Google Scholar

[18]

R. Zweimüller, Mixing limit theorems for ergodic transformations, J. Theor. Prob., 20 (2007) 1059-1071. doi: 10.1007/s10959-007-0085-y.  Google Scholar

[19]

A. Zygmund, "Trigonometric Series: Vols. I, II," Second edition, Cambridge University Press, London-New York, 1968.  Google Scholar

show all references

References:
[1]

P. Billingsley, "Convergence of Probability Measures," Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 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) 290-294. 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), 1-34. 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), 653-660.  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), 479-492. doi: 10.1017/S0143385701001237.  Google Scholar

[7]

A. M. Garsia, "Topics in Almost Everywhere Convergence," Lectures in Advanced Mathematics, 4, Markham Publishing Co., Chicago, Ill., 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), 73-98.  Google Scholar

[9]

P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application," Probability and Mathematical Statistics, Academic Press, Inc., New York-London, 1980.  Google Scholar

[10]

C. Hoffman, A Markov random field which is $K$ but not Bernoulli, Israel J. Math., 112 (1999), 249-269. 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), 349-362.  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., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970.  Google Scholar

[13]

M. Peligrad and C. Peligrad, On the invariance principle under martingale approximation, Stoch. Dyn., 11 (2011), 95-105. 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), 2009-2022. doi: 10.1214/10-AOP530.  Google Scholar

[15]

V. A. Rohlin, Lectures on the entropy of measure-preserving transformations, Russian Mathematical Surveys, 22 (1967), 1-52. Google Scholar

[16]

K. Schmidt, "Dynamical Systems of Algebraic Origin," Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[17]

K. Schmidt, On joint recurrence, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 837-842. doi: 10.1016/S0764-4442(99)80115-3.  Google Scholar

[18]

R. Zweimüller, Mixing limit theorems for ergodic transformations, J. Theor. Prob., 20 (2007) 1059-1071. doi: 10.1007/s10959-007-0085-y.  Google Scholar

[19]

A. Zygmund, "Trigonometric Series: Vols. I, II," Second edition, Cambridge University Press, London-New York, 1968.  Google Scholar

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