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Let $X \sim \operatorname{Binom}[(n,p)]$ and $Y \sim \operatorname{Poisson}[f(X)]$, where f is a convex function. Are there any good tail bounds for $Y$? For instance, are there any Chernoff-style bounds for $Y$?

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    The article "The moment bound is tighter than Chernoff's bound for positive tail probabilities", referenced at [this post](http://math.stackexchange.com/questions/66403/about-bound-based-on-r-th-central-moment) may be of interest.2011-11-07

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