poisson-binomial mixture tail bound 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 $Y$?
 A: The Cheroff's bound, for positive random variable, and arbitrary $\theta$ such that the moment generating function $\mathcal{M}_Y(\theta)$ exists:
$$
  \mathbb{P}(Y \ge t) \le \mathrm{e}^{-\theta t} \mathcal{M}_Y(\theta)   =  \mathbb{E} \left( \mathrm{e}^{-\theta t + f(X) \left( \mathrm{e}^\theta - 1 \right))}  \right)
$$
If $g_\theta(X) = \mathrm{e}^{-\theta t + f(X) \left( \mathrm{e}^\theta - 1 \right))}$ happens to be concave, then by Jensen's inequality we would have:
$$
   \mathbb{P}(Y \ge  t) \le  \mathrm{e}^{-\theta t + f( \mathbb{E} \left(X\right)) \left( \mathrm{e}^\theta - 1 \right))}  = \mathrm{e}^{-\theta t + f( n p ) \left( \mathrm{e}^\theta - 1 \right))} 
$$
The inequality above would be true for any such $\theta$ that $g_\theta^{\prime\prime}(X) \le 0$ for all $X \ge 0$:
$$
 0 \ge  g_\theta^{\prime\prime}(x) = g(x) \left\{ \left( f^\prime(x) ( \mathrm{e}^{\theta} - 1) \right)^2 + f^{\prime\prime}(x) ( \mathrm{e}^{\theta} - 1)  \right\}
$$
Alternatively one could look at the moment bound:
$$
  \mathbb{P}(Y > t) \le \frac{1}{t} \mathbb{E}(Y) = \frac{1}{t} \mathbb{E}(f(X))
$$
Again, if $f(x)$ is concave, by Jensen's inequality we would have
$$
    \mathbb{P}(Y > t) \le \frac{f(n p)}{t}
$$
