  We consider the empirical versions of geometric quantile and halfspace depth,
and study their extremal behaviour as a function of the sample size. The
objective of this study is to establish connection between the rates of
convergence and tail behaviour of the corresponding underlying distributions.
The intricate interplay between the sample size and the parameter driving the
extremal behaviour forms the main result of this analysis. In the process, we
also fill certain gaps in the understanding of population versions of geometric
quantile and halfspace depth.
