Half-region depth for stochastic processes

We study the concept of half-region depth, introduced in López-Pintado and Romo (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 n level for many smoothed processes.