It is shown that under appropriate conditions the $s$th cumulant of a von Mises statistic or a $U$ (or $V$) statistic is $O(n^{-s + 1}), s \geq 2$, as the sample size $n$ goes to infinity. A possible route toward the derivation of an asymptotic expansion of the characteristic function is indicated.
@article{1176993600,
author = {Bhattacharya, R. N. and Puri, M. L.},
title = {On the Order of Magnitude of Cumulants of Von Mises Functionals and Related Statistics},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 346-354},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993600}
}
Bhattacharya, R. N.; Puri, M. L. On the Order of Magnitude of Cumulants of Von Mises Functionals and Related Statistics. Ann. Probab., Tome 11 (1983) no. 4, pp. 346-354. http://gdmltest.u-ga.fr/item/1176993600/