The Annals of Probability
- Ann. Probab.
- Volume 42, Number 2 (2014), 794-817.
Komlós–Major–Tusnády approximation under dependence
István Berkes, Weidong Liu, and Wei Biao Wu
Full-text: Open access
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.
Article information
Source
Ann. Probab., Volume 42, Number 2 (2014), 794-817.
Dates
First available in Project Euclid: 24 February 2014
Permanent link to this document
https://projecteuclid.org/euclid.aop/1393251303
Digital Object Identifier
doi:10.1214/13-AOP850
Mathematical Reviews number (MathSciNet)
MR3178474
Zentralblatt MATH identifier
1308.60037
Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes 60G17: Sample path properties
Keywords
Stationary processes strong invariance principle KMT approximation weak dependence nonlinear time series ergodic sums
Citation
Berkes, István; Liu, Weidong; Wu, Wei Biao. Komlós–Major–Tusnády approximation under dependence. Ann. Probab. 42 (2014), no. 2, 794--817. doi:10.1214/13-AOP850. https://projecteuclid.org/euclid.aop/1393251303
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