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

Let T : X -> X be an Axiom A diffeomorphism, m the Gibbs state for a Holder continuous function g. Assume that f : X -> R-d is a Holder continuous function with integral x fdm = 0. If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix sigma(2) := sigma(2)(f) such that S-f/root n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix sigma(2). Moreover, there exists a real number A > 0 such that, for any integer n >= 1, Pi (m(*) (1/root nS(n)f), N (0, sigma(2))) <= A/root n where m(*) (1/root nS(n)(f)) denotes the distribution of 1/root nS(n)(f) with respect to in, and Pi is the Prokhorov metric.