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

• 成果类型: Article
• 署名作者: Pène, F
• 署名单位: Universite de Bretagne Occidentale
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000036
• 发表日期: 2004
• 页码: 2477-2525
• 关键词: dynamical-systems; FLOWS; clt

摘要

The motivation of this work is the study of the error term e(t)(epsilon)(x,omega) in the averaging method for differential equations perturbed by a dynamical system. Results of convergence in distribution for ( e(t)(epsilon) (x,.)/ rootepsilon) epsilon>0 have been established in Khas'minskii [Theory Probab. Appl. 11 (1966) 211-228], Kifer [Ergodic Theory Dynamical Systems 15 (1995) 1143-1172] and Pene [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 (e(t)(epsilon) (x,.)/rootepsilon) epsilon> 0 when epsilon goes to 0 (t > 0 and x is an element of 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+alpha)) for all alpha > 0 (for the Prokhorov metric). This result can be seen as an extension of the main result of Pene [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 (e(t)(epsilon)(x,.)/rootepsilon) epsilon > 0 with rate of convergence in O(epsilon(1/2-alpha)) (for the Prokhorov metric). We close this paper with a discussion (in the Appendix) about the behavior of the quantity parallel to sup(0less than or equal totless than or equal toT0) \e(t)(epsilon) (x,.)\infinityparallel to L-P under less stringent hypotheses.