We determine the scaling properties of the probability distribution of the smoothed density field in N-body simulations of expanding universes with scale-free initial power-spectra, {\delta(k)\(2)] proportional to k(n), with particular attention to the predictions of the stable clustering hypothesis. We concentrate our analysis on the ratios S-Q(l) = xi(Q)/xi(2)(Q-1), where xi(Q) is the average Q-body correlation function over a cell of a radius l. According to the stable clustering hypothesis, S-Q should not depend on scale. We performed measurements for Q less than or equal to 5. The behavior of the higher order correlations is studied through that of the void probability distribution, P-0(l), which is the probability of finding an empty cell of radius l. If the stable clustering hypothesis applies, the function P-0 should also exhibit remarkable scaling properties. In our analysis, we take carefully into account various misleading effects, such as initial grid contamination, loss of dynamics due to the short-range softening of the forces, and finite volume size of our simulations. Only after correcting for the latter do we find agreement of the measured S-Q with the expected self-similar behavior. Otherwise, P-0 is only weakly sensitive to such effects and closely follows the expected self-similar behavior. The ratios exhibit two plateaus separated by a smooth transition when xi(2)(l) similar to 1. In the weakly nonlinear regime xi(2) less than or similar to 1, the results are in reasonable agreement with the predictions of perturbation theory. In the strongly nonlinear regime, xi(2) > 1, the S-Q values are larger than in the weakly nonlinear regime, and increasingly so with -n. They are well fitted by the expression S-Q = (xi(2)/100)(0.04(Q-2))S-Q for all n and 3 less than or equal to Q less than or equal to 5. This weak dependence on scale proves a small but significant departure from the stable clustering predictions, at least for n = 0 and n = +1. The analysis P-0 confirms that the expected scale invariance of the functions S-Q is not exactly attained in the part of the nonlinear regime we probe, except possibly for n = -2 and marginally for n = -1. In these two cases, our measurements are not accurate enough to be discriminant. On the other hand, we could demonstrate that the observed power-law behavior of S-Q cannot be generalized as such to arbitrary order in Q. Indeed, we show that this would induce scaling properties of P-0 that are incompatible with those measured.
