## The Annals of Statistics

### No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions

#### 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 $\inf_n \sup_A |\mu_n(A) - \mu(A)| \geq \frac{1}{2} - \delta \quad\text{almost surely}.$

#### Article information

Source
Ann. Statist., Volume 18, Number 3 (1990), 1496-1499.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347765

Digital Object Identifier
doi:10.1214/aos/1176347765

Mathematical Reviews number (MathSciNet)
MR1062724

Zentralblatt MATH identifier
0707.60026

JSTOR