On the Order of Magnitude of Cumulants of Von Mises Functionals and Related Statistics

Abstract

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.