A multidimensional central limit theorem with speed of convergence for axiom a diffeomorphisms

Abstract

Let T : X → X be an Axiom A diffeomorphism, m the Gibbs state for a Hölder continuous function g. Assume that f : X → ℝ d is a Hölder continuous function with ∫ X fdm = 0. If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ 2 :=σ 2( 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 σ 2. Moreover, there exists a real number A > 0 such that, for any integer n ≥ 1, II ( m * ( 1 n S n f ) , N ( 0 , σ 2 ) ) ≤ A n , where m * ( 1 n S n f ) denotes the distribution of 1 n S n f with respect to m, and II is the Prokhorov metric.