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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2$\begingroup$ 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. $\endgroup$– Anton GeraschenkoCommented Dec 11, 2009 at 1:51
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$\begingroup$ Are you sure you meant from the point of view of statistics, rather than probability? What does that exactly mean? $\endgroup$– Ori Gurel-GurevichCommented Dec 11, 2009 at 4:37
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$\begingroup$ 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. $\endgroup$– Chris ConidisCommented Dec 11, 2009 at 11:44
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1$\begingroup$ Probability and statistics are actually quite different fields. In the former you begin with a mathematical model and prove theorems about its behavior. In the latter you look at some data and try to produce a model with similar behavior. So it would be good to be clear about what you mean here. $\endgroup$– Nate EldredgeCommented Sep 29, 2010 at 16:32
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5 Answers
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I don't know about statistics per se but the best introduction to martingales period is Williams' Probability with Martingales:
http://www.amazon.com/Probability-Martingales-Cambridge-Mathematical-Textbooks/dp/0521406056
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$\begingroup$ I'll second that. Excellent book. (I have taught from it.) $\endgroup$ Commented Dec 11, 2009 at 1:48
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$\begingroup$ I'll third that. Excellent book. (I have learned from it.) $\endgroup$ Commented Jan 5, 2010 at 13:45
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$\begingroup$ For a more statistics-y text, go to Williams' "Weighing the odds". $\endgroup$ Commented Jan 5, 2010 at 13:46
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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: http://www.probabilityandfinance.com/chapters/toc.html
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1$\begingroup$ This book and approach are very interesting, IMO. $\endgroup$– R HahnCommented Sep 29, 2010 at 17:01
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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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$\begingroup$ 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? $\endgroup$ Commented Dec 11, 2009 at 8:54
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$\begingroup$ @Yemon: He doesn't have the reputation to comment yet. $\endgroup$ Commented Dec 11, 2009 at 13:18
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Have a look at "The Doctrine of Chances: Probabilistic Aspects of Gambling" by Stewart Ethier http://www.springer.com/mathematics/probability/book/978-3-540-78782-2
Lots of examples related to just about every casino game you've ever heard of, and a few you have not.