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 Hölder continuous function g. Assume that f : X → ℝd is a Hölder continuous function with ∫Xfdm = 0. If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ2 :=σ2(f  ) such that Snfn 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, II(m*(1nSnf),N(0,σ2)) ≤ An,where m*(1nSnf) denotes the distribution of 1nSnf with respect to m, and II is the Prokhorov metric.