Multiple decorrelation and rate of convergence in multidimensional limit theorems for the Prokhorov metric
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Françoise Pène [1]
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[1] University of Brest
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 3, 2, 2004, págs. 2477-2525
- Idioma: inglés
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Resumen
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.
