  The statistical meaning of the local non-Gaussianity parameters f_NL and g_NL
is examined in detail. Their relations to the skewness and the kurtosis of the
probability distribution of density fluctuations are shown to obey simple
fitting formulae, accurate on galaxy-cluster scales. We argue that the
knowledge of f_NL and g_NL is insufficient for reconstructing a well-defined
distribution of density fluctuations. However, by weakening the statistical
significance of f_NL and g_NL, it is possible to reconstruct a well-defined pdf
by using a truncated Edgeworth series. We give some general guidelines on the
use of such a series, noting in particular that 1) the Edgeworth series cannot
represent models with nonzero f_NL, unless g_NL is nonzero also, 2) the series
cannot represent models with g_NL<0, unless some higher-order non-Gaussianities
are known. Finally, we apply the Edgeworth series to calculate the effects of
g_NL on the abundances of massive clusters and large voids. We show that the
abundance of voids may generally be more sensitive to high-order
non-Gaussianities than the cluster abundance.
