Abstract
The empirical distribution function $P_n$ converges with probability 1 to a true distribution $P$ in $R^k$, uniformly over measurable convex sets, if and only if $P$ is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.
Citation
W. F. Eddy. J. A. Hartigan. "Uniform Convergence of the Empirical Distribution Function Over Convex Sets." Ann. Statist. 5 (2) 370 - 374, March, 1977. https://doi.org/10.1214/aos/1176343801
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