  We present a detailed comparison of the predictions of perturbation theory
for the averaged J-point correlation functions, $\xibar_J$, with the results of
numerical simulations of gravitational clustering. We have carried out a
systematic analysis of this method using ensembles of simulations with
different numbers of particles, different box sizes and using different
particle arrangements and clustering amplitudes in the initial conditions. We
estimate $\xibar_J$, for $J=2-10$, from moments of counts-in-cells. We find
significant non-linear effects in the variance, $J=2$, even at scales as large
as $R \sim 30 \Mpc$. Perturbation theory gives remarkable agreement at large
scales, where $\xibar_2 \simlt 1$, with the measured hierarchical amplitudes
$S_J=\xibar_J/\xibar_2^{J-1}$. We have followed the evolution of $\xibar_J$ in
time and find that for a change in the effective redshift of at least $\Delta z
\simeq 2$ the amplitudes $S_J$ remain unchanged, despite the fact that the
$\xibar_J$ have evolved by large factors, $\simeq 10^{J-1}$. We illustrate how
these results can be applied to interpret the clustering in galaxy surveys and
conclude that the observed hierarchical pattern in the APM is compatible with
gravitational evolution in unbiased, initially Gaussian, models.
