On the measure of anchored Gaussian simplices, with applications to multivariate medians

We consider anchored Gaussian (cid:96) -simplices in the d -dimensional Euclidean space, that is, simplices with one fixed vertex y ∈ R d and the remaining vertices X 1 , . . . , X (cid:96) randomly sampled from the d -variate standard normal distribution. We determine the distribution of the measure of such simplices for any d , any (cid:96) , and any anchor point y , which is of interest, e.g., when studying the asymptotics of U-statistics based on such simplex measures. We provide two proofs of the results. The first one is short but is not self-contained as it crucially relies on a technical result for non-central Wishart distributions. The second one is a simple and self-contained proof, that also provides some geometric insight on the results. Quite nicely, variations on this second argument reveal intriguing distributional identities on products of central and non-central chi-square distributions with Beta-distributed non-centrality parameters. We independently establish these distributional identities by making use of Mellin transforms. Beyond the aforementioned use to study the asymptotics of some U-statistics, our results do find natural applications in the context of robust location estimation, as we illustrate by considering a class of simplex-based multivariate medians that contains the celebrated spatial median and Oja median as special cases. Throughout, our results are confirmed by numerical experiments.