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.
I don't know about statistics per se but the best introduction to martingales period is Williams' Probability with Martingales:
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".
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
Shiryaev's "Probability" is an excellent source.
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.