  We study the concept of half-region depth, introduced by Lopez-Pintado and
Romo in 2011. We show that for a wide variety of standard stochastic processes,
such as Brownian motion and other symmetric stable processes with stationary
independent increments tied down at 0, half-region depth assigns depth zero to
all sample functions. To alleviate this difficulty we introduce a method of
smoothing, which often not only eliminates the problem of zero depth, but
allows us to extend the theoretical results on consistency in that paper up to
the $\sqrt n$ level for many smoothed processes.
