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March 2014 Komlós–Major–Tusnády approximation under dependence
István Berkes, Weidong Liu, Wei Biao Wu
Ann. Probab. 42(2): 794-817 (March 2014). DOI: 10.1214/13-AOP850


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.


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István Berkes. Weidong Liu. Wei Biao Wu. "Komlós–Major–Tusnády approximation under dependence." Ann. Probab. 42 (2) 794 - 817, March 2014.


Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 1308.60037
MathSciNet: MR3178474
Digital Object Identifier: 10.1214/13-AOP850

Primary: 60F17 , 60G10 , 60G17

Keywords: ergodic sums , KMT approximation , nonlinear time series , Stationary processes , Strong invariance principle , Weak dependence

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 2 • March 2014
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