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

Abstract

The motivation of this work is the study of the error term etɛ(x,ω) in the averaging method for differential equations perturbed by a dynamical system. Results of convergence in distribution for $({\frac{e_{t}^{\varepsilon}(x,\cdot)}{\sqrt{\varepsilon}}})_{\varepsilon>0}$ have been established in Khas’minskii [Theory Probab. Appl. 11 (1966) 211–228], Kifer [Ergodic Theory Dynamical Systems 15 (1995) 1143–1172] and Pène [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 $({\frac{e_{t}^{\varepsilon}(x,\cdot)}{\sqrt{\varepsilon}}})_{\varepsilon>0}$ when ɛ goes to 0 (t>0 and $x\in{\bf R}^{d}$ 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+α) for all α>0 (for the Prokhorov metric). This result can be seen as an extension of the main result of Pène [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 $({\frac{e_{t}^{\varepsilon}(x,\cdot)}{\sqrt{\varepsilon}}})_{\varepsilon>0}$ with rate of convergence in O(ɛ1/2−α) (for the Prokhorov metric). We close this paper with a discussion (in the Appendix) about the behavior of the quantity $\Vert\sup_{0\le t\le T_{0}}\vert e_{t}^{\varepsilon}(x,\cdot)\vert_{\infty}\Vert_{L^{p}}$ under less stringent hypotheses.