  Using a non-perturbative method developed in a previous work (paper II), we
derive the probability distribution $P(\delta_R)$ of the density contrast
within spherical cells in the quasi-linear regime for some non-Gaussian initial
conditions. We describe three such models. The first one is a straightforward
generalization of the Gaussian scenario. It can be seen as a phenomenological
description of a density field where the tails of the linear density contrast
distribution would be of the form $P_L(\delta_L) \sim
e^{-|\delta_L|^{-\alpha}}$, where $\alpha$ is no longer restricted to 2 (as in
the Gaussian case). We derive exact results for $P(\delta_R)$ in the
quasi-linear limit. The second model is a physically motivated isocurvature CDM
scenario. Our approach needs to be adapted to this specific case and in order
to get convenient analytical results we introduce a simple approximation (which
is not related to the gravitational dynamics but to the initial conditions).
Then, we find a good agreement with the available results from numerical
simulations for the pdf of the linear density contrast for $\delta_{L,R} \ga
0$. We can expect a similar accuracy for the non-linear pdf $P(\delta_R)$.
Finally, the third model corresponds to the small deviations from Gaussianity
which arise in standard slow-roll inflation. We obtain exact results for the
pdf of the density field in the quasi-linear limit, to first-order over the
primordial deviations from Gaussianity.
