We control parameter distortion in the generalized renormalization procedure provided a certain set of starting conditions is satisfied. This allows us to prove that for a $C^\infty$--open set of unimodal families, almost all parameters inside an interval present either stochastic dynamics or a renormalization (in the classical sense). Moreover, easy consequences are that renormalization happens densely on this interval and stochastic behaviour with positive measure. A wide range use of this approach would rely mostly on proving that the starting conditions are satisfied for general families.