Split invariance principles for stationary processes.

By István Berkes, Siegfried Hormann, Johannes Schauer. 2011

46 citations recorded by the source corpus.

The results of Komlos, Major and Tusnady 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, $2 0$, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249–280] removed the logarithmic factor, reaching the Komlos–Major–Tusnady 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.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/46b8e7451e4a81783aa211c56bdfb138b2d6790a