Multiple decorrelation and rate of convergence in multidimensional limit theorems for the Prokhorov metric

The motivation of this work is the study of the error term e_t^{\epsilon}(x,\omega) in the averaging method for differential equations perturbed by a dynamical system. Results of convergence in distribution for (\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} have been established in Khas'minskii [Theory Probab. Appl. 11 (1966) 211-228], Kifer [Ergodic Theory Dynamical Systems 15 (1995) 1143-1172] and P\`ene [ESAIM Probab. Statist. 6 (2002) 33-88]. We are interested here in the question of the rate of convergence in distribution of the family of random variables (\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} when \epsilon goes to 0 (t>0 and x\inR^d being fixed). We will make an assumption of multiple decorrelation property (satisfied in several situations). We start by establishing a simpler result: the rate of convergence in the central limit theorem for regular multidimensional functions. In this context, we prove a result of convergence in distribution with rate of convergence in O(n^{-1/2+\alpha}) for all \alpha>0 (for the Prokhorov metric). This result can be seen as an extension of the main result of P\`ene [Comm. Math. Phys. 225 (2002) 91-119] to the case of d-dimensional functions. In a second time, we use the same method to establish a result of convergence in distribution for (\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} with rate of convergence in O(\epsilon^{1/2-\alpha}) (for the Prokhorov metric).