# Polynomial Markov versus Chernoff Bound for random variables

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?

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

• Thank you for letting me have better understanding of this Taylor expansion trick! Jan 12 at 20:21