  We study the distribution of a general class of asymptoticallylinear
statistics which are symmetric functions of $N$ independent observations. The
distribution functions of these statistics are approximated by an Edgeworth
expansion with a remainder of order $o(N^{-1})$. The Edgeworth expansion is
based on Hoeffding's decomposition which provides a stochastic expansion into a
linear part, a quadratic part as well as smaller higher order parts. The
validity of this Edgeworth expansion is proved under Cram\'er's condition on
the linear part, moment assumptions for all parts of the statistic and an
optimal dimensionality requirement for the non linear part.
