Asymptotic normality of trimmed means in higher dimensions (Q1110210)

X\({}_ 1,...,X_ n\) are i.i.d. d-dimensional random vectors with common distribution function F. For each nonnegative y, S(y) is a bounded measurable subset of d-dimensional space, with a bounded measurable boundary, with \(S(y_ 1)\subseteq S(y_ 2)\) if \(y_ 1\leq y_ 2\). For \(r\geq 1\), let \(y_{n-r}\) denote inf [y\(>0:\) S(y) contains exactly n-r of \(X_ 1,...,X_ n]\). Then define \(X_ n^{(1)}=X_{i_ n(1)}\), where \(i_ n(1)=i\) such that \(X_ i\not\in S(y_{n-r})\), \(i\neq i_ n(1),...,i_ n(r-1)\), \(2\leq r\leq n\). Define trimmed sums by \[ (r)_{S_ n}=X_ n^{(n)}+...+X_ n^{(r+1)},\quad 1\leq r<n. \] It is shown that if r is the largest integer in na, where \(0<a<1\), the trimmed sum is asymptotically normal when normed and centered appropriately, without requirement of moment or other conditions on the tail of F, but with certain smoothness conditions on F. The rate of convergence of the approach to normality is studied.