A variety of estimators for the parameters of the Generalized Pareto distribution, the approximating distribution for excesses over a high threshold, have been proposed, always assuming the underlying data to be independent. We recently proved that the likelihood moment estimators are consistent estimators for the parameters of the Generalized Pareto distribution for the case where the underlying data arises from a (stationary) linear process with heavy-tailed innovations. In this paper we derive the bivariate asymptotic normality under some additional assumptions and give an explicit example on how to check these conditions by using asymptotic expansions.