On the consistency of the spacings test for multivariate uniformity

Original · EN

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ Rᵈ that is based on the maximal spacing generated by i.i.d. points X₁,,Xₙ in K, i.e., the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the case d > 1 are only availabe under uniformity, a key element of the proof is a suitable coupling.