A set of observations is partitioned into $k$ clusters by optimizing a clustering criterion $W$. The asymptotic distribution of this clustering criterion may be determined simply in certain cases where the optimal sample partition differs negligibly from the optimal population partition. Detailed proofs are given in the one-dimensional case when the clustering criterion to be minimized is within cluster sum of squares. The asymptotic distributions are used to compute approximate significance levels of tests for the presence of clusters, and of tests for bimodality.
"Asymptotic Distributions for Clustering Criteria." Ann. Statist. 6 (1) 117 - 131, January, 1978. https://doi.org/10.1214/aos/1176344071