The Annals of Probability

Split invariance principles for stationary processes

István Berkes, Siegfried Hörmann, and Johannes Schauer

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Abstract

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.

Article information

Source
Ann. Probab., Volume 39, Number 6 (2011), 2441-2473.

Dates
First available in Project Euclid: 17 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1321539125

Digital Object Identifier
doi:10.1214/10-AOP603

Mathematical Reviews number (MathSciNet)
MR2932673

Zentralblatt MATH identifier
1236.60037

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

Keywords
Stationary processes strong invariance principle KMT approximation dependence increments of partial sums

Citation

Berkes, István; Hörmann, Siegfried; Schauer, Johannes. Split invariance principles for stationary processes. Ann. Probab. 39 (2011), no. 6, 2441--2473. doi:10.1214/10-AOP603. https://projecteuclid.org/euclid.aop/1321539125


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