A Central Limit Theorem for k-Means Clustering

A set of n points in Euclidean space is partitioned into the k groups that minimize the within groups sum of squares. Under the assumption that the n points come from independent sampling on a fixed distribution, conditions are found to assure asymptotic normality of the vector of means of the k groups. The method of proof makes novel application of a functional central limit theorem for empirical processes--a generalization of Donsker's theorem due to Dudley.