Komlós-Major-Tusnády approximation under dependence

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.

Published in at http://dx.doi.org/10.1214/13-AOP850 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Asymptotic spectral theory for nonlinear time series
• Split invariance principles for stationary processes

Cited by in corpus (7)

• Berry-Esseen theorems under weak dependence
• Central limit theorems for high dimensional dependent data
• Rates in almost sure invariance principle for dynamical systems with some hyperbolicity
• Backward CUSUM for Testing and Monitoring Structural Change with an Application to COVID-19 Pandemic Data
• Scaling Limits and Generic Bounds for Exploration Processes
• An improvement of Tusnády's inequality in the bulk
• An alternative to the coupling of Berkes-Liu-Wu for strong approximations