The empirical spatial distribution and the plug-in spatial depth are uniformly L1-consistent over all of R^d with a dimension-free rate of 1/sqrt(n).
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Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator
The empirical spatial distribution and the plug-in spatial depth are uniformly L1-consistent over all of R^d with a dimension-free rate of 1/sqrt(n).