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2012

On the Convergence of the Empirical Distribution

Berend, Daniel, Kontorovich, Aryeh

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We develop a general technique for bounding the tail of the total variation distance between the empirical and the true distributions over countable sets.

Our methods sharpen a deviation bound of Devroye (1983) for distributions over finite sets, and also hold for the broader class of distributions with countable support.
We also provide some lower bounds of possible independent interest.

On the Convergence of the Empirical Distribution · Around

Built on

Complete convergence and the law of large numbers
Pao-Lu Hsu and Herbert Robbins · 1947
Earlier work this paper cites.
Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator
Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz · 1956
Earlier work this paper cites.
Mathematics of Statistics, 3rd ed
John F. Kenney and Ernest S. Keeping · 1962
Earlier work this paper cites.
The equivalence of weak, strong and complete convergence in L 1 L_{1} for kernel density estimates
Luc Devroye · 1983
Earlier work this paper cites.

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Then

Combinatorial methods in density estimation
Luc Devroye and Gábor Lugosi · 2001
Later among the works it cites.
On choosing and bounding probability metrics
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Elements of information theory
Thomas M. Cover and Joy A. Thomas · 2006
Later among the works it cites.

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