Constructive Proofs of Concentration Bounds

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Constructive Proofs of Concentration Bounds
We give a simple combinatorial proof of the Chernoff-Hoeffding concentration bound [Che52, Hoe63], which says that the sum of independent {0, 1}-valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, but rather uses a simple and intuitive counting argument. In addition, our proof is constructive in the following sense: if the sum of the given random variables is not concentrated around the expectation, then we can efficiently find (with high probability) a subset of the random variables that are statistically dependent. As simple corollaries, we also get the concentration bounds for [0, 1]-valued random variables and Azuma's inequality for martingales [Azu67]. We interpret the Chernoff-Hoeffding bound as a statement about Direct Product Theorems. Informally, a Direct Product Theorem says that the complexity of solving all k instances of a hard problem increases exponentially with k...
Russell Impagliazzo, Valentine Kabanets
Added 26 Oct 2010
Updated 26 Oct 2010
Type Conference
Year 2010
Authors Russell Impagliazzo, Valentine Kabanets
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