Split invariance principles for stationary processes

TL;DR

This paper introduces a novel approach to Wiener approximation for stationary processes, achieving near-optimal rates for a broader range of moments and dependence structures, with applications in limit theorems and statistical testing.

Contribution

It extends Wiener approximation techniques by allowing a second component, enabling near-optimal rates for all moments greater than two in dependent processes.

Findings

Achieves near $o(n^{1/p})$ rates for all $p>2$ using a second Wiener component.

Applies to wide classes of linear, nonlinear, and dynamical systems.

Enables new limit theorems and statistical tests for stationary processes.

Abstract

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, $2<p\le4$, with error term $o(n^{1/p}(\log n)^{\gamma})$, $\gamma>0$, 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.