The comment at the end about the moment bounds being at least as good as the Chernoff-type bounds is actually a really simple idea. Say we’re working with non-negative random variables. Recall the Chernoff uses something like

We could have instead used the “best” moment bound

The claim (from this 1995 paper by Philips and Nelson) is

This means whatever bound we could prove by taking the moment-generating function (on the left), it will never be better than the best moment bound.

Why is the claim true? Suppose the RHS minimum is . Then , for all . And hence, for any ,

Which means the LHS infimum is also at least . Easy! The paper extends this to general random variables (using a simple conditioning trick), and also gives other results, including examples where the moment bound can be much better.

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