Asymptotic expansions in functional limit theorems

Asymptotic expansions for a class of functional limit theorems are investigated. It is shown that the expansions in this class fit into a common scheme, defined by a sequence of functions hn ([var epsilon]1,..., [var epsilon]n), n >= 1, of "weights" (for nobservations), which are smooth, symmetric, compatible and have vanishing first derivatives at zero. Then hn(n-1/2,..., n-1/2) admits an asymptotic expansion in powers of n-1/2. Applications to quadratic von Mises functionals, the C.L.T. in Banach spaces, and the invariance principle are discussed.