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July 2004 Multiple decorrelation and rate of convergence in multidimensional limit theorems for the Prokhorov metric
Françoise Pène
Ann. Probab. 32(3B): 2477-2525 (July 2004). DOI: 10.1214/009117904000000036

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 O1/2−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.

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Françoise Pène. "Multiple decorrelation and rate of convergence in multidimensional limit theorems for the Prokhorov metric." Ann. Probab. 32 (3B) 2477 - 2525, July 2004. https://doi.org/10.1214/009117904000000036

Information

Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1055.60015
MathSciNet: MR2078548
Digital Object Identifier: 10.1214/009117904000000036

Subjects:
Primary: 37D , 60F05

Keywords: averaging method , dynamical systems , hyperbolic systems , multidimensional limit theorems , multiple decorrelation , Prokhorov metric , Speed of convergence

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3B • July 2004
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