  The spatial distribution has been widely used to develop various
nonparametric procedures for finite dimensional multivariate data. In this
paper, we investigate the concept of spatial distribution for data in infinite
dimensional Banach spaces. Many technical difficulties are encountered in such
spaces that are primarily due to the noncompactness of the closed unit ball. In
this work, we prove some Glivenko-Cantelli and Donsker-type results for the
empirical spatial distribution process in infinite dimensional spaces. The
spatial quantiles in such spaces can be obtained by inverting the spatial
distribution function. A Bahadur-type asymptotic linear representation and the
associated weak convergence results for the sample spatial quantiles in
infinite dimensional spaces are derived. A study of the asymptotic efficiency
of the sample spatial median relative to the sample mean is carried out for
some standard probability distributions in function spaces. The spatial
distribution can be used to define the spatial depth in infinite dimensional
Banach spaces, and we study the asymptotic properties of the empirical spatial
depth in such spaces. We also demonstrate the spatial quantiles and the spatial
depth using some real and simulated functional data.

Methodology