Limit law for some modified ergodic sums

An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.