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The motivation of this work is the study of the error terme_t^{\epsilon}(x,\omega) in the averaging method for differential equationsperturbed by a dynamical system. Results of convergence in distribution for(\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} have beenestablished in Khas'minskii [Theory Probab. Appl. 11 (1966) 211-228], Kifer[Ergodic Theory Dynamical Systems 15 (1995) 1143-1172] and P\`ene [ESAIMProbab. Statist. 6 (2002) 33-88]. We are interested here in the question of therate of convergence in distribution of the family of random variables(\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} when \epsilon goesto 0 (t>0 and x\inR^d being fixed). We will make an assumption of multipledecorrelation property (satisfied in several situations). We start byestablishing a simpler result: the rate of convergence in the central limittheorem for regular multidimensional functions. In this context, we prove aresult of convergence in distribution with rate of convergence inO(n^{-1/2+\alpha}) for all \alpha>0 (for the Prokhorov metric). This result canbe 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 usethe same method to establish a result of convergence in distribution for(\frac{e_t^{\epsilon}(x,\cdot)}{\sqrt\epsilon})_{\epsilon>0} with rate ofconvergence in O(\epsilon^{1/2-\alpha}) (for the Prokhorov metric).

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