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Suppose that $X\geq0$, and that the moment generating function of $X$ exists in an interval around 0. Given some $\delta>0$ and integer $k=1,2,...$, show that $$\inf_{k=0,1,...}\frac{E(|X|^k)}{\delta^k} \leq \inf_{\lambda>0} \frac{E(e^{\lambda X})}{e^{\lambda \delta}}. $$

Consequently, an optimized bound based on polynomial moments is always at least as good as the Chernoff upper bound. Could anyone enlighten me how to prove this?

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Let $b$ denote the LHS. Expanding $e^{\lambda X}$ in a power series you can deduce that $$E(e^{\lambda X}) \ge \sum_{k \ge 0} \frac {b \lambda^k \delta^k}{k!}=b e^{\lambda \delta} \,.$$

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  • $\begingroup$ Thank you for letting me have better understanding of this Taylor expansion trick! $\endgroup$
    – Hepdrey
    Jan 12 at 20:21

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