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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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