Asymptotics for a class of dependent random variables

We consider a class of dependent random variables where the dependence structure involves a factor driven by Sn/nSn/n. Under very mild conditions for the innovation, we obtain several asymptotic results for the partial sums including basic asymptotics and strong limit theorems. The fact that the asymptotic properties differ strikingly in a neighbour of a critical point makes the model very interesting.