An invariance principle for weakly dependent stationary general models

TL;DR

This paper refines a weak invariance principle for stationary sequences, establishing convergence rates under weaker dependence conditions and moment assumptions than previous works, especially for non-causal models.

Contribution

It introduces a weaker dependence condition, called lambda, and extends the invariance principle to non-causal models with moments greater than two.

Findings

Establishes a weak invariance principle under lambda dependence.

Achieves convergence rates in the CLT with moments > 2.

Extends results to non-causal stationary sequences.

Abstract

The aim of this article is to refine a weak invariance principle for stationary sequences given by Doukhan & Louhichi (1999). Since our conditions are not causal our assumptions need to be stronger than the mixing and causal $\theta$-weak dependence assumptions used in Dedecker & Doukhan (2003). Here, if moments of order $>2$ exist, a weak invariance principle and convergence rates in the CLT are obtained; Doukhan & Louhichi (1999) assumed the existence of moments with order $>4$. Besides the previously used $\eta$- and $\kappa$-weak dependence conditions, we introduce a weaker one, $\lambda$, which fits the Bernoulli shifts with dependent inputs.