  We apply the time-renormalization group approach to study the effect of
primordial non-Gaussianities in the non-linear evolution of cosmological dark
matter density perturbations. This method improves the standard perturbation
approach by solving renormalization group-like equations governing the dynamics
of gravitational instability. The primordial bispectra constructed from the
dark matter density contrast and the velocity fields represent initial
conditions for the renormalization group flow. We consider local, equilateral
and folded shapes for the initial non-Gaussianity and analyze as well the case
in which the non-linear parameter f_{NL} parametrizing the strength of the
non-Gaussianity depends on the momenta in Fourier space through a power-law
relation, the so-called running non-Gaussianity. For the local model of
non-Gaussianity we compare our findings for the power-spectrum with those of
recent N-body simulations and find that they accurately fit the N-body data up
to wave-numbers k \sim 0.25 h/Mpc at z=0. We also present predictions for the
(reduced) matter bispectra for the various shapes of non-Gaussianity.
