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 $\inf_n \sup_A |\mu_n(A) - \mu(A)| \geq \frac{1}{2} - \delta \quad\text{almost surely}.$
Publié le : 1990-09-14
Classification:
Empirical measure,
total variation distance,
singular continuous distributions,
62G05,
60E05
@article{1176347765,
author = {Devroye, Luc and Gyorfi, Laszlo},
title = {No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions},
journal = {Ann. Statist.},
volume = {18},
number = {1},
year = {1990},
pages = { 1496-1499},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347765}
}
Devroye, Luc; Gyorfi, Laszlo. No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions. Ann. Statist., Tome 18 (1990) no. 1, pp. 1496-1499. http://gdmltest.u-ga.fr/item/1176347765/