Asymptotics for multivariate trimming (Q1193405)

A multivariate \(\alpha\)-trimming procedure, which consists in intersecting all closed halfspaces containing \(100(1-\alpha)\%\) of the sample \(\xi_ 1,\dots,\xi_ n\) taken from a distribution \(P\) on \(R^ d\), or more, is considered. Let \(C_ n\) and \(C_ \alpha\) be the intersections of all closed halfspaces with \(P_ n\)-measure and \(P\)- measure \(1-\alpha\) or greater, respectively, and \[ r_ n(u)=\inf\{r\geq 0: ru\not\in C_ n\},\quad r_ \alpha(u)=\inf\{r\geq 0: ru\not\in C_ \alpha\} \] be the radius functions on the unit sphere. The standardized sample radius function \(r_ n(r_ n-r_ \alpha)\) is proved to converge weakly as \(n\to\infty\) to a Gaussian process. The limiting distribution for the normalized unit vector normal to the boundary is established.