On moment conditions for valid formal Edgeworth expansions (Q1120213)

The validity of formal Edgeworth expansions for statistics of the form \(W_ n=\sqrt{n\{H(\bar Z)-H(\mu)\}}\) was established by the authors [Ann. Stat. 6, 434-451 (1978; Zbl 0396.62010)] under moment conditions which are sometimes too severe, where \(\bar Z=(1/n)\sum^{n}_{j=1}Z_ j\) is the mean of i.i.d. vectors and H is a smooth function in a neighbourhood of \(\mu =E(Z_ j)\). If all the components of grad H(\(\mu)\) are nonzero then one cannot significantly weaken these established moment conditions. In this article the authors provide a relaxation of the moment conditions in case grad H(\(\mu)\) has some zero components. Then, they sketch a method for obtaining an asymptotic expansion of a conditional distribution, where the conditioning is with respect to the coordinates of \(Z_ j\) for which the corresponding elements of grad H(\(\mu)\) are equal to zero.