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Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots of examples. Thanks.

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Community wiki! Go! – Harry Gindi Dec 11 '09 at 1:18
By the way, the downvote was not me. I flagged this for community wiki though. So don't gather to lynch me just yet. – Harry Gindi Dec 11 '09 at 1:23
I've converted this question to wiki, but please make such questions community wiki in the future. The general rule is that if it wouldn't make sense for people to earn reputation for their answers (as in the case of references, where the reference should really be getting the reputation, rather than the person who answered), then the question should be community wiki. – Anton Geraschenko Dec 11 '09 at 1:51
Are you sure you meant from the point of view of statistics, rather than probability? What does that exactly mean? – Ori Gurel-Gurevich Dec 11 '09 at 4:37
Yes, by statistics I meant probability. I just want the standard definition of martingale, because I'm coming at these things from the point of view of computability theory and algorithmic randomness, where the definition that we use (I am pretty sure) is slightly different than the standard one used in probability theory. Thanks to everyone who has replied/commented so far. – Chris Conidis Dec 11 '09 at 11:44
up vote 10 down vote accepted

I don't know about statistics per se but the best introduction to martingales period is Williams' Probability with Martingales:

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I'll second that. Excellent book. (I have taught from it.) – Harald Hanche-Olsen Dec 11 '09 at 1:48
I'll third that. Excellent book. (I have learned from it.) – Kevin O'Bryant Jan 5 '10 at 13:45
For a more statistics-y text, go to Williams' "Weighing the odds". – Kevin O'Bryant Jan 5 '10 at 13:46

no doubt that Probability with Martingales is excellent, but I am not sure that this is a very satisfying book for someone interested in "the point of view of statistics".

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Fair point, although it's not clear to me just what the original poster meant by "the point of view of statistics" - hopefully this will be clarified. BTW, your answer should perhaps be a comment on Steve's rather than a separate one? – Yemon Choi Dec 11 '09 at 8:54
@Yemon: He doesn't have the reputation to comment yet. – Harald Hanche-Olsen Dec 11 '09 at 13:18

I haven't read it, but I saw Glenn Shafer give a talk on his and Vladimir Vovk's game theoretical approach to probability. It could be an interesting/more intuitive approach than the usual measure theoretic approaches. They discuss martingales in several chapters of their book and I'm certain there's lots of applications/examples especially from finance. Might be worth a look:

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This book and approach are very interesting, IMO. – R Hahn Sep 29 '10 at 17:01

Shiryaev's "Probability" is an excellent source.

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Have a look at "The Doctrine of Chances: Probabilistic Aspects of Gambling" by Stewart Ethier

Lots of examples related to just about every casino game you've ever heard of, and a few you have not.

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