**Doob’s Inequality **

Let be a (discrete) non-negative sub-martingale, bounded in i.e. with for some fixed . Define by , and .

Then . Further, .

*Proof:* Fix and define the stopping time . Then which is -measurable and thus by The Stopping Time Lemma since

Also by Fubini

Using both of these facts and Fubini again

Hölder’s inequality states that for then . Now which is in hence . Thus

Hence from above

If then

If then the inequality holds trivially.

Finally,

and now apply monotone convergence to obtain

Advertisements