## No empirical probability measure can converge in the total variation sense for all distributions.(English)Zbl 0707.60026

Summary: 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 $\inf_{n} \sup_{A}| \mu_ n(A)-\mu (A)| \geq -\delta \quad almost\quad surely.$

### MSC:

 60E99 Distribution theory 62G05 Nonparametric estimation
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