Work

Komlós–Major–Tusnády approximation under dependence

Year: 2014

Type: article

Abstract: The celebrated results of Komlós, Major and Tusnády [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.

Source: The Annals of Probability

DOI: 10.1214/13-aop850

Authors: István Berkes, Weidong Liu, Wei Biao Wu

Institutions: Shanghai Jiao Tong University, Graz University of Technology, University of Chicago

Language: English

Topic: Probability and Statistical Research

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: National Natural Science Foundation of China, Austrian Science Fund, Hungarian Scientific Research Fund

Awards: G3323208847, G6937649604, Probabilistic analysis of multivariate problems