Open Access
November 2011 Split invariance principles for stationary processes
István Berkes, Siegfried Hörmann, Johannes Schauer
Ann. Probab. 39(6): 2441-2473 (November 2011). DOI: 10.1214/10-AOP603


The results of Komlós, Major and Tusnády give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294–2320] obtained Wiener approximation of a class of dependent stationary processes with finite pth moments, 2 < p ≤ 4, with error term o(n1/p(log n)γ), γ > 0, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249–280] removed the logarithmic factor, reaching the Komlós–Major–Tusnády bound o(n1/p). No similar results exist for p > 4, and in fact, no existing method for dependent approximation yields an a.s. rate better than o(n1/4). In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to o(n1/p) for arbitrary p > 2. This extends the scope of applications of the results essentially, as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term (epidemic) changes in stationary processes. Our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems.


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István Berkes. Siegfried Hörmann. Johannes Schauer. "Split invariance principles for stationary processes." Ann. Probab. 39 (6) 2441 - 2473, November 2011.


Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1236.60037
MathSciNet: MR2932673
Digital Object Identifier: 10.1214/10-AOP603

Primary: 60F17 , 60G10 , 60G17

Keywords: Dependence , increments of partial sums , KMT approximation , Stationary processes , Strong invariance principle

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • November 2011
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