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

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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.

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

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