Acta mathematica scientia, Series B >
A MULTIDIMENSIONAL CENTRAL LIMIT THEOREM WITH SPEED OF CONVERGENCE FOR AXIOM A DIFFEOMORPHISMS
Received date: 2008-09-19
Revised date: 2010-03-01
Online 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.
Let T: X→ X be an Axiom A diffeomorphism, m the Gibbs state for a H\"older continuous function g. Assume that f: X→ Rd 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 Snf/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 Snf ), N(0, σ2))≤A /√n,
where m* (1/√n Snf) denotes the distribution of 1/√n Snf with respect to m, and ∏ is the Prokhorov metric.
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 . DOI: 10.1016/S0252-9602(11)60303-2
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