Hoeffding's lemma


In probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable. It is named after the Finnish-American mathematical statistician Wassily Hoeffding.
The proof of Hoeffding's lemma uses Taylor's theorem and Jensen's inequality. Hoeffding's lemma is itself used in the proof of McDiarmid's inequality.

Statement of the lemma

Let X be any real-valued random variable with expected value, such that almost surely, i.e. with probability one. Then, for all,
Note that the proof is based on the assumption that the random variable has zero expectation, hence the and in the lemma must satisfy. For any random variable, which do not obey this assumption we can define, which obey the assumptions and apply the proof on.

A brief proof of the lemma

Since is a convex function of, we have
So,
Let, and
Then, since
Taking derivative of,
By Taylor's expansion,
Hence,

More detailed proof

First note that if one of or is zero, then and the inequality follows. If both are nonzero, then must be negative and must be positive.
Next, recall that is a convex function on the real line:
Applying to both sides of the above inequality gives us:
Let and define:
is well defined on, to see this we calculate:
The definition of implies
By Taylor's theorem, for every real there exists a between and such that
Note that:
Therefore,
This implies