The Annals of Probability

Komlós–Major–Tusnády approximation under dependence

István Berkes, Weidong Liu, and Wei Biao Wu

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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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Ann. Probab., Volume 42, Number 2 (2014), 794-817.

First available in Project Euclid: 24 February 2014

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Primary: 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes 60G17: Sample path properties

Stationary processes strong invariance principle KMT approximation weak dependence nonlinear time series ergodic sums


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.

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