多维Prokhorov度量极限定理的多重去相关和收敛速率
Multiple decorrelation and rate of convergence in multidimensional limit theorems for the Prokhorov metric
摘要：
本文的目的是研究受动力系统扰动的微分方程的平均方法中的误差项e_t^ε（x，ω）。关于（\frac{e_t^ε(x,\cdot)}{\sqrtε}）_{ε>0}依分布收敛的结果在Khas'minskii [Theory Probab. 11（1966）211-228]、Kifer [遍历理论动力系统15（1995）1143-1172]和Pène [ESAIM Probab.中央集权主义者6（2002）33-88]。本文研究随机变量族（\frac{e_t^ε(x,\cdot)}{\sqrtε}）_{ε>0}在ε趋于0（t>0，x \inR^d固定）时分布的收敛速率问题。我们将假设多重去相关属性（在几种情况下都满足）。我们首先建立一个简单的结果：正则多维函数的中心极限定理的收敛速率。在这方面，我们证明了一个结果，收敛速率为O（n^{-1/2+α}）的依分布收敛，所有α>0（对于Prokhorov度量）。这个结果可以看作是Pène [Comm.Math.Phys.225（2002）91-119]的主要结果到d维函数情况的推广。第二次，我们用同样的方法建立了（\frac{e_t^ε(x,\cdot)}{\sqrtε}）_{ε>0}依分布收敛的一个结果，其收敛速率为O（ε^{1/2-α}）（对于Prokhorov度量）.
英文摘要：
The motivation of this work is the study of the error term e_t^ε(x,ω) in the averaging method for differential equations perturbed by a dynamical system. Results of convergence in distribution for (\frac{e_t^ε(x,\cdot)}{\sqrtε})_{ε>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^ε(x,\cdot)}{\sqrtε})_{ε>0} when εgoes 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+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^ε(x,\cdot)}{\sqrtε})_{ε>0} with rate of convergence in O(ε^{1/2-α}) (for the Prokhorov metric).
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