We discuss the non--linear growth of the excess kurtosis parameter of the smoothed density fluctuation field $��$, $S_4\equiv[\lan��^{\,4}\ran-3\lan��^{\,2}\ran^2]/ \lan��^{\,2}\ran^3$ in an Einstein--de Sitter universe. We assume Gaussian primordial density fluctuations with scale--free power spectrum $P(k)\propto k^{\,n}$ and analyze the dependence of $S_4$ on primordial spectral index $n$, after smoothing with a Gaussian filter. As already known for the skewness ratio $S_3$, the kurtosis parameter is a {\it decreasing function} of $n$, both in exact perturbative theory and in the Zel'dovich approximation. The parameter $S_4$ provides a powerful statistics to test different cosmological scenarios.

11 pages in Latex (plus 1 figure), SISSA 127/93/A