KOMLOS-MAJOR-TUSNADY APPROXIMATION UNDER DEPENDENCE

• 成果类型: Article
• 署名作者: Berkes, Istvan; Liu, Weidong; Wu, Wei Biao
• 署名单位: Graz University of Technology; Shanghai Jiao Tong University; University of Chicago
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP850
• 发表日期: 2014
• 页码: 794-817
• 关键词: CENTRAL-LIMIT-THEOREM; partial-sums; INVARIANCE-PRINCIPLES

摘要

The celebrated results of Komlos, Major and Tusnady [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] give optimal Wiener approximation for the partial sums of i.i.d. random variables and provide a powerful tool in probability and statistics. In this paper we extend KMT approximation for a large class of dependent stationary processes, solving a long standing open problem in probability theory. Under the framework of stationary causal processes and functional dependence measures of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150-14154], we show that, under natural moment conditions, the partial sum processes can be approximated by Wiener process with an optimal rate. Our dependence conditions are mild and easily verifiable. The results are applied to ergodic sums, as well as to nonlinear time series and Volterra processes, an important class of nonlinear processes.