A MULTIDIMENSIONAL CENTRAL LIMIT THEOREM WITH SPEED OF CONVERGENCE FOR AXIOM A DIFFEOMORPHISMS

doi:10.1016/S0252-9602(11)60303-2

Acta mathematica scientia,Series B ›› 2011, Vol. 31 ›› Issue (3): 1123-1132.doi: 10.1016/S0252-9602(11)60303-2

• Articles • Previous Articles Next Articles

A MULTIDIMENSIONAL CENTRAL LIMIT THEOREM WITH SPEED OF CONVERGENCE FOR AXIOM A DIFFEOMORPHISMS

XIA Hong-Qiang, SHAN Da-Yao

College of Science, Wuhan Textile University, Wuhan 430073, China； Department of Mathematics and Computer Science, Qinzhou University, Qinzhou 535000, China

Received:2008-09-19 Revised:2010-03-01 Online:2011-05-20 Published:2011-05-20
Supported by:
This work is partially supported by the National Natural Science Foundation of China (10571174) and the Scientific Research Foundation of Ministry of Education for Returned Overseas Chinese Scholars and Scientific Research Foundation of Ministry of Human Resources and Social Security for Returned Overseas Chinese Scholars.

Abstract

Abstract:

Let T: X→ X be an Axiom A diffeomorphism, m the Gibbs state for a H\"older continuous function g. Assume that f: X→ R^d is a H\"older
continuous function with ∫_Xf dm=0. If the components of f are cohomologously independent, then there exists a positive definite
symmetric matrix σ²:=σ²(f) such that S_n^f/n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ². Moreover, there exists a real number A>0 such that, for any integer n≥1,

∏(m_*{1/√n S_n^f), N(0, σ²))≤A /√n,
where m_*(1/√n S_n^f) denotes the distribution of 1/√n S_n^f with respect to m, and ∏ is the Prokhorov metric.

Key words: multidimensional central limit theorem, Axiom A diffeomorphisms, symbolic dynamics, transfer operator

CLC Number:

37A25

XIA Hong-Qiang, SHAN Da-Yao. A MULTIDIMENSIONAL CENTRAL LIMIT THEOREM WITH SPEED OF CONVERGENCE FOR AXIOM A DIFFEOMORPHISMS[J].Acta mathematica scientia,Series B, 2011, 31(3): 1123-1132.

Trendmd

References

[1] Bowen R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin, New York: Springer-Verlag, 1975

[2] Viana M. Stochastic Dynamics of Deterministic Systems. Lecture Notes XXI Braz Math Colloq, IMPA: Rio de Janeiro, 1997

[3] Gouëzel S. Berry-Esseen theorem and local limit theorem for non uniformly expanding maps. Annales de l'IHP Probabilités et
Statistiques, 2005, 6: 997--1024

[4] Young L S. Statistical properties of dynamical systems with some hyperbolicity. Ann Math, 1998, 2: 585--650

[5] Xia H. Local central limit theorem and Berry-Essen theorem for some non-uniformly hyperbolic diffeomorphisms. Acta Mathematica Scientia, 2010, 30(3): 701--712

[6] Dolgopyat D. Limit theorems for partially hyperbolic systems. Trans Amer Math Soc, 2004, 4: 1637--1689

[7] Melbourne I, Nicol M. Almost sure invariant principle for nonuniformly hyperbolic systems. Comm Math Phys, 2005, 1: 131--146

[8] Liverani C. Central limit theorem for deterministic systems//Ledrappier F, Lewowicz J, Newhouse S, eds. International Conference on Dynamical Systems Pitman Research Notes in Math 362. Harlow: Longman Group Ltd, 1996: 56--75

[9] Hennion H, Herv\'e L. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness.
Lecture Notes in Math, New York: Springer, 2001

[10] Fan A, Jiang Y. Spectral theory of transfer operators//Jiang Y, Wang Y. Complex Dynamics and Related Topics. New Studies in Advanced Mathematics, Vol 5. International Press, 2004: 63--128

[11] Melbourne I, Nicol M. A vector-valued almost sure invariant principle for hyperbolic dynamical systems. Ann Probab, 2009, 2: 478--505

[12] Xia H, Zhang Z. Multidimensional central limit theorem with speed of convergence for uniformly expanding maps. Journal of
Huazhong Normal University, 2008, 3: 323--327

[13] Pène F. Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen and to the Sinai billiard. Ann Appl Probab, 2005, 4: 2331--2392

[14] Walters P. An Introduction to Ergodic Theory. New York: Springer Verlag, 1982

[15] Parry W, Pollicott M. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Ast\'erisque, 1990: 187--188

[16] Zinsmeister M. Thermodynamic Formalism and Holomorphic Dynamical Systems. SMF/AMS Texts and Monographs, Vol. 2, Providence RI: American Mathematical Society, 2000

[17] Dudley M. Real Analysis and Probability. Wadsworth and Brooks Cole, CA: Pacific Grove, 1989

[18] Yurinskii V. A smoothing inequality for estimates of the Levy-Prokhorov distance. Theory Probab Appl, 1975, 1: 1--10

A MULTIDIMENSIONAL CENTRAL LIMIT THEOREM WITH SPEED OF CONVERGENCE FOR AXIOM A DIFFEOMORPHISMS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 2

Metrics

Comments

Recommended 10

[1]	Peng SUN. A GENERALIZATION OF GAUSS-KUZMIN-LÉVY THEOREM [J]. Acta mathematica scientia,Series B, 2018, 38(3): 965-972.
[2]	Xia Hongqiang; Ge Yunfei; Zhang Zhengjie. EQUIDISTRIBUTION OF CLOSED ORBITS OF BILLIARD FLOWS ON THE PLANE [J]. Acta mathematica scientia,Series B, 2007, 27(3): 465-472.