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It is known that the optional stopping theorem from martingale theory is a very powerful theorem in probability theory in statistics.

I have heard of a probability course at Stanford where martingales and optional stopping is introduced at the beginning, then much of the course material is derived from this principle.

Unfortunately there are no course notes. Does anyone know where I can find these kind of derivations, or for instance, whether it is possible to derive concentration bounds like Azuma's inequality from OST?

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Maybe you will be interested in David Williams's book Probability with Martingales, which is intended as a textbook for a first course. It does take the approach of proving many classical results (strong law of large numbers, etc) using martingale techniques. I don't believe that it gets as far as concentration bounds, however.

If you can find out who taught the course at Stanford, it may be worth asking them if they have any notes / references / materials they would be willing to share.

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  • $\begingroup$ I have a chance to look at the course material mentioned by OP, personally I strongly dislike that approach because that makes students think that everything (they encountered) is a martingale w.r.t. some process and (wrongly) believe that conditional expectation is no more than a sequence of operators. In fact, such an approach eliminate all exchangeability elements which should be a part of prob. theory. But William's book is a nice balance! good answer! $\endgroup$
    – Henry.L
    Apr 12, 2017 at 20:21

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