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August  2020, 40(8): 4665-4687. doi: 10.3934/dcds.2020197

Fluctuations of ergodic sums on periodic orbits under specification

Manfred Denker 1, , Samuel Senti 2, and  Xuan Zhang 3,

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-909, Brazil
3. 

Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil

Received  April 2019 Published  May 2020

We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.

Keywords: Central limit theorem, Lindeberg condition, dynamical array, specification, measure of maximal entropy, mixture of normal distributions.
Mathematics Subject Classification: Primary: 37A50; Secondary: 37B99, 60F05.
Citation: Manfred Denker, Samuel Senti, Xuan Zhang. Fluctuations of ergodic sums on periodic orbits under specification. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4665-4687. doi: 10.3934/dcds.2020197
References:
[1]

J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.  Google Scholar

[2]

J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.  doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[4]

R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[5]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279.  Google Scholar

[6]

R. C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, UT, 2007.  Google Scholar

[7]

R. Burton and M. Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc., 302 (1987), 715-726.  doi: 10.1090/S0002-9947-1987-0891642-6.  Google Scholar

[8]

M. Denker, The central limit theorem for dynamical systems, in Dynamical Systems and Ergodic Theory, Banach Center Publ., 23, PWN, Warsaw, 1989, 33–62.  Google Scholar

[9]

M. Denker, J. Duan and M. McCourt, Pseudorandom numbers for conformal measures, Dyn. Syst., 24 (2009), 439-457.  doi: 10.1080/14689360903002019.  Google Scholar

[10]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364.  Google Scholar

[11]

M. Denker and G. Keller, On $U$-statistics and v. Mises' statistics for weakly dependent processes, Z. Wahrsch. Verw. Gebiete, 64 (1983), 505-522.  doi: 10.1007/BF00534953.  Google Scholar

[12]

M. Denker, S. Senti and X. Zhang, The Lindeberg theorem for Gibbs-Markov dynamics, Nonlinearity, 30 (2017), 4587-4613.  doi: 10.1088/1361-6544/aa8ca2.  Google Scholar

[13]

P. Doukhan, Mixing. Properties and Examples, Lecture Notes in Statistics, 85, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2642-0.  Google Scholar

[14]

M. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR, 188 (1969), 739-741.   Google Scholar

[15]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields, 128 (2004), 82-122.  doi: 10.1007/s00440-003-0300-4.  Google Scholar

[16]

S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electron. J. Probab., 19 (2014), 30pp. doi: 10.1214/EJP.v19-3427.  Google Scholar

[17]

B. Hasselblatt, Introduction to hyperbolic dynamics and ergodic theory, in Ergodic Theory and Negative Curvature, Lecture Notes in Math, 2164, Springer, Cham, 2017, 1–124. doi: 10.1007/978-3-319-43059-1_1.  Google Scholar

[18]

N. Haydn, M. Nicol, S. Vaienti and L. Zhang, Central limit theorems for the shrinking target problem, J. Stat. Phys., 153 (2013), 864-887.  doi: 10.1007/s10955-013-0860-3.  Google Scholar

[19]

H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874.  Google Scholar

[20]

I. A. Ibragimov and Y. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971,473pp.  Google Scholar

[21]

A. Klenke, Probability Theory, Universitext, Springer, London, 2014. doi: 10.1007/978-1-4471-5361-0.  Google Scholar

[22]

D. Kwietniak, M. Łącka and P. Oprocha, A panorama of specification-like properties and their consequences, in Dynamics and Numbers, Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016,155–186. doi: 10.1090/conm/669/13428.  Google Scholar

[23]

C. Liverani, Central limit theorem for deterministic systems, in International Conference on Dynamical Systems), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996, 56–75.  Google Scholar

[24]

R. Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems, 5 (1985), 71-88.  doi: 10.1017/S0143385700002765.  Google Scholar

[25]

V. V. Petrov, Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer-Verlag, New York-Heidelberg, 1975. doi: 10.1007/978-3-642-65809-9.  Google Scholar

[26]

J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab., 11 (1983), 772-788.  doi: 10.1214/aop/1176993522.  Google Scholar

[27]

D. Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity, 5 (1992), 1223-1236.  doi: 10.1088/0951-7715/5/6/002.  Google Scholar

[28]

K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.  Google Scholar

[29]

D. Thomine, A generalized central limit theorem in infinite ergodic theory, Probab. Theory Related Fields, 158 (2014), 597-636.  doi: 10.1007/s00440-013-0491-2.  Google Scholar

[30]

D. Thomine, Variations on a central limit theorem in infinite ergodic theory, Ergodic Theory Dynam. Systems, 35 (2015), 1610-1657.  doi: 10.1017/etds.2013.114.  Google Scholar

show all references

References:
[1]

J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.  Google Scholar

[2]

J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.  doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[4]

R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[5]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/BFb0081279.  Google Scholar

[6]

R. C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, UT, 2007.  Google Scholar

[7]

R. Burton and M. Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc., 302 (1987), 715-726.  doi: 10.1090/S0002-9947-1987-0891642-6.  Google Scholar

[8]

M. Denker, The central limit theorem for dynamical systems, in Dynamical Systems and Ergodic Theory, Banach Center Publ., 23, PWN, Warsaw, 1989, 33–62.  Google Scholar

[9]

M. Denker, J. Duan and M. McCourt, Pseudorandom numbers for conformal measures, Dyn. Syst., 24 (2009), 439-457.  doi: 10.1080/14689360903002019.  Google Scholar

[10]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/BFb0082364.  Google Scholar

[11]

M. Denker and G. Keller, On $U$-statistics and v. Mises' statistics for weakly dependent processes, Z. Wahrsch. Verw. Gebiete, 64 (1983), 505-522.  doi: 10.1007/BF00534953.  Google Scholar

[12]

M. Denker, S. Senti and X. Zhang, The Lindeberg theorem for Gibbs-Markov dynamics, Nonlinearity, 30 (2017), 4587-4613.  doi: 10.1088/1361-6544/aa8ca2.  Google Scholar

[13]

P. Doukhan, Mixing. Properties and Examples, Lecture Notes in Statistics, 85, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2642-0.  Google Scholar

[14]

M. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR, 188 (1969), 739-741.   Google Scholar

[15]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields, 128 (2004), 82-122.  doi: 10.1007/s00440-003-0300-4.  Google Scholar

[16]

S. Gouëzel and I. Melbourne, Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electron. J. Probab., 19 (2014), 30pp. doi: 10.1214/EJP.v19-3427.  Google Scholar

[17]

B. Hasselblatt, Introduction to hyperbolic dynamics and ergodic theory, in Ergodic Theory and Negative Curvature, Lecture Notes in Math, 2164, Springer, Cham, 2017, 1–124. doi: 10.1007/978-3-319-43059-1_1.  Google Scholar

[18]

N. Haydn, M. Nicol, S. Vaienti and L. Zhang, Central limit theorems for the shrinking target problem, J. Stat. Phys., 153 (2013), 864-887.  doi: 10.1007/s10955-013-0860-3.  Google Scholar

[19]

H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874.  Google Scholar

[20]

I. A. Ibragimov and Y. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971,473pp.  Google Scholar

[21]

A. Klenke, Probability Theory, Universitext, Springer, London, 2014. doi: 10.1007/978-1-4471-5361-0.  Google Scholar

[22]

D. Kwietniak, M. Łącka and P. Oprocha, A panorama of specification-like properties and their consequences, in Dynamics and Numbers, Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016,155–186. doi: 10.1090/conm/669/13428.  Google Scholar

[23]

C. Liverani, Central limit theorem for deterministic systems, in International Conference on Dynamical Systems), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996, 56–75.  Google Scholar

[24]

R. Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems, 5 (1985), 71-88.  doi: 10.1017/S0143385700002765.  Google Scholar

[25]

V. V. Petrov, Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer-Verlag, New York-Heidelberg, 1975. doi: 10.1007/978-3-642-65809-9.  Google Scholar

[26]

J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab., 11 (1983), 772-788.  doi: 10.1214/aop/1176993522.  Google Scholar

[27]

D. Ruelle, Thermodynamic formalism for maps satisfying positive expansiveness and specification, Nonlinearity, 5 (1992), 1223-1236.  doi: 10.1088/0951-7715/5/6/002.  Google Scholar

[28]

K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.  Google Scholar

[29]

D. Thomine, A generalized central limit theorem in infinite ergodic theory, Probab. Theory Related Fields, 158 (2014), 597-636.  doi: 10.1007/s00440-013-0491-2.  Google Scholar

[30]

D. Thomine, Variations on a central limit theorem in infinite ergodic theory, Ergodic Theory Dynam. Systems, 35 (2015), 1610-1657.  doi: 10.1017/etds.2013.114.  Google Scholar

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