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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 and 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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[24]

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

[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.

[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.

[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.

[28]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[24]

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

[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.

[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.

[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.

[28]

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

[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.

[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.

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