Edgeworth expansions for L-statistics

We study the approximation by a short Edgeworth expansion of the distribution function of normalized linear combinations [formula] of order statistics of n independent random variables with common distribution function F. Under the assumptions [formula] for some p1, q1, r1 ∈ R p2, q2, r2, C≥0, к ∈ [0.5/4) with an appropriate balance in these parameters, and under additional moment conditions, the rate of uniform convergence is shown to be of order n -1. Moreover, a special case is considered where the cjn are generated by a sequence of weight functions of a special structure.