  The concept of data depth leads to a center-outward ordering of multivariate
data, and it has been effectively used for developing various data analytic
tools. While different notions of depth were originally developed for finite
dimensional data, there have been some recent attempts to develop depth
functions for data in infinite dimensional spaces. In this paper, we consider
some notions of depth in infinite dimensional spaces and study their properties
under various stochastic models. Our analysis shows that some of the depth
functions available in the literature have degenerate behaviour for some
commonly used probability distributions in infinite dimensional spaces of
sequences and functions. As a consequence, they are not very useful for the
analysis of data satisfying such infinite dimensional probability models.
However, some modified versions of those depth functions as well as an infinite
dimensional extension of the spatial depth do not suffer from such degeneracy,
and can be conveniently used for analyzing infinite dimensional data.
