Split invariance principles for stationary processes

Details Split invariance principles for stationary processes Split invariance principles for stationary processes

2012-02-13

arxiv.org/abs/1202.2640v1

Annals of Probability 2011, Vol. 39, No. 6, 2441-2473

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP603 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP603

The results of Koml\'{o}s, Major and Tusn\'{a}dy 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 $p$th moments, $20$, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249--280] removed the logarithmic factor, reaching the Koml\'{o}s--Major--Tusn\'{a}dy bound $o(n^{1/p})$. No similar results exist for $p>4$, and in fact, no existing method for dependent approximation yields an a.s. rate better than $o(n^{1/4})$. In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to $o(n^{1/p})$ 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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