Edgeworth approximations for distributions of symmetric statistics

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.