Independence of Four Projective Criteria for the Weak Invariance Principle

Olivier Durieu (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

Let $(X_i)_{i\in\Z}$ be a regular stationary process for a given filtration. The weak invariance principle holds under the condition $\sum_{i\in\Z}\|P_0(X_i)\|_2<\infty$ (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)).

Domaines: Probabilités [math.PR]

Olivier Durieu. Independence of Four Projective Criteria for the Weak Invariance Principle. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2009, 5, pp.21-27. ⟨hal-00291499⟩