Hoeffding's lemma

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In probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable.[1] It is named after the FinnishAmerican 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[edit]

Let X be any real-valued random variable with expected value and such that almost surely. Then, for all ,

Note that because of the assumption that the random variable has zero expectation, the and in the lemma must satisfy .

A brief proof of the lemma[edit]

Since is a convex function of , we have


Let , and

Then, since

Taking derivative of ,

for all h.

By Taylor's expansion,


(The proof below is the same proof with more explanation.)

More detailed proof[edit]

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:


This implies

See also[edit]


  1. ^ Pascal Massart (26 April 2007). Concentration Inequalities and Model Selection: Ecole d'Eté de Probabilités de Saint-Flour XXXIII - 2003. Springer. p. 21. ISBN 978-3-540-48503-2.