Independence of Four Projective Criteria for the Weak Invariance Principle

Abstract

Let $(X_i{)}_{i\in \mathbb{Z}}$ be a regular stationary process for a given filtration. The weak invariance principle holds under the condition ${\sum }_{i\in \mathbb{Z}}\Vert P_0(X_i){\Vert }_2<\infty$ (see Hannan (1979)}, Dedecker and Merlevède (2003), Deddecker, Merlevéde and Volný (2007)). In this paper, we show that this criterion is independent of other known criteria: the martingale-coboundary decomposition of Gordin (see Gordin (1969, 1973)), the criterion of Dedecker and Rio (see Dedecker and Rio (2000)) and the condition of Maxwell and Woodroofe (see Maxwell and Woodroofe (2000), Peligrade and Utev (2005), Volný (2006, 2007)).

arXiv comment6 pages

References in corpus (1)

• Comparison between criteria leading to the weak invariance principle

Cited by in corpus (7)

• On martingale approximations and the quenched weak invariance principle
• Invariance principle via orthomartingale approximation
• Limit theorems under the Maxwell-Woodroofe condition in Banach spaces
• H{ö}lderian weak invariance principle under Maxwell and Woodroofe condition
• Quenched Invariance Principles for the Discrete Fourier Transforms of a Stationary Process
• On the functional CLT for stationary Markov Chains started at a point
• An invariance principle for stationary random fields under Hannan's condition