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

The spatial distribution in infinite dimensional spaces and related quantiles and depths