Asymptotic Normality of Trimmed Means in Higher Dimensions

Abstract

A representation for the distribution of the trimmed sum of vector-valued random variables is obtained, generalising a one-dimensional formula. The trimming is with respect to observations falling outside a fixed family of sets, e.g., spheres. Asymptotic normality of the heavily trimmed sum, when normed and centered in different ways, is proved, and rates of convergence are given for some cases.