Convergence of quantile and depth regions

Since contours of multi-dimensional depth functions often characterize the distribution, it has become of interest to consider structural properties and limit theorems for the sample contours (see Zuo and Serfling (2000)). For finite dimensional data Massé and Theodorescu (1994) [14] and Kong and Mizera (2012) have made connections of directional quantile envelopes to level sets of half-space (Tukey) depth. In the recent paper (Kuelbs and Zinn, 2014) we showed that half-space depth regions determined by evaluation maps of a stochastic process are not only uniquely determined by related upper and lower quantile functions for the process, but limit theorems have also been obtained. In this paper we study the consequences of these results when applied to finite dimensional data in greater detail. The methods we employ here are based on Kuelbs and Zinn (2015) and Kuelbs and Zinn (2013).