Structural properties and convergence results for contours of sample statistical depth functions

Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are studied. Structural properties of the regions enclosed by contours, such as affine equivariance, nestedness, connectedness and compactness, and almost sure convergence results for sample depth contours, are established. Also, specialized results are established for some popular depth functions, including halfspace depth, and for the case of elliptical distributions. Finally, some needed foundational results on almost sure convergence of sample depth functions are provided.