  The computation of the Tukey depth, also called halfspace depth, is very
demanding, even in low dimensional spaces, because it requires the
consideration of all possible one-dimensional projections. In this paper we
propose a random depth which approximates the Tukey depth. It only takes into
account a finite number of one-dimensional projections which are chosen at
random. Thus, this random depth requires a very small computation time even in
high dimensional spaces. Moreover, it is easily extended to cover the
functional framework.
  We present some simulations indicating how many projections should be
considered depending on the sample size and on the dimension of the sample
space. We also compare this depth with some others proposed in the literature.
It is noteworthy that the random depth, based on a very low number of
projections, obtains results very similar to those obtained with other depths.
