  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.
