Abstract
For any sequence of empirical probability measures $\{\mu_n\}$ on the Borel sets of the real line and any $\delta > 0$, there exists a singular continuous probability measure $\mu$ such that $\inf_n \sup_A |\mu_n(A) - \mu(A)| \geq \frac{1}{2} - \delta \quad\text{almost surely}.$
Citation
Luc Devroye. Laszlo Gyorfi. "No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions." Ann. Statist. 18 (3) 1496 - 1499, September, 1990. https://doi.org/10.1214/aos/1176347765
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