Bennett bounds for sparse binomial fluctuations
= Bennett bounds for sparse binomial fluctuations
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For a <binomial distribution> with $p_n\to0$ and $np_n\to\infty$, the <Bennett inequality> at $C$ standard deviations tends to $2e^{-C^2/2}$, whereas the <Hoeffding inequality> tends to two. The raw <Bennett inequality> bound is eventually smaller for any fixed $C>0$; after clipping at one it is nontrivial in the limit only when $C>\sqrt{2\log2}$.