Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding.

Though martingales is a very well explored area of probability and I have seen it come up in a course on randomized algorithms, I kept feeling that it required more deeper thinking than most other things in the course. As in, I would time and again have difficulties articulating to myself why is this an interesting mathematical object to look at or maybe more precisely, I was not able to convince myself how martingales will arise in ordinary ruminations of abstract thought. (I know it did arise in "some ruminations", but so far I have been unable to see a clear motivation behind these objects).

I have seen a few applications -- like they can be used to get concentration inequalities even when your random variables are dependent (and the absolute deltas are bounded). I have also seen the gambler motivation; that did not reduce my uneasiness with martingales much :(

I guess what I would like is the following. I have given a few lectures on (basic) applications of probabilistic method and I have a good feel for this. But I cannot say the same for martingales. As of now, it seems like an alien mathematical object to me and I certainly cannot imagine myself talking about martingales yet. Please let me know if you have some suggestions/reading material which can help with this. Also, I guess I would be perfectly happy with some more approachable/playful applications of martingales.

Thanks!

Solution 1:

As an applied probabilist, I've had similar issues with martingales. From my own questions, it seems that they are useful for mostly theoretical reasons to prove that certain properties hold for the process under study. If you can identify a process as a martingale, then you know some rather general things about that process that may lead you to being able to get specific answers to your questions.

The generality and abstractness of martingales means that any process whose mean varies monotonically in time can potentially be converted into a martingale by adding/subtracting the proper conditional expected value. You can then apply martingale theorems (optional stopping, convergence) which, if you are lucky, will lead to explicit answers.

Therefore, if you approach martingales from the point of view of "how can I calculate numbers using these", you may be rather disappointed, as there are only a few such instances I've run across.

However, if you approach it from the point of view "Ah...process $X_t$ is a martingale, hence any non-clarvoyant stopping strategy will NOT affect its expected value". Now, this realization is useful for an applied mathematician/probabilist and could potentially save you a lot of time (not to mention embarrassment) trying clever (but ultimately futile) ways to determine how to optimize the expected value of such a process. That's been my major take-away. If you're a theoretician, then they are just plain awesome due to the great results you can prove for processes that are martingales :-)

Solution 2:

From statistics view point, we also use martingale to show properties of some statistical test. The most important example is to show the optimal property of likelihood ratio test (LR) when given both null and alternative simple hypothesis. By using martingale, one could show that LR is martingale under null hypothesis and sub-martingale under alternative hypothesis. This result shows LR test is optimal which is another way to show Neyman-Pearson theorem. And it is very amazing in cases of simple hypothesis under null hypothesis sample size changes won't affect the test statistics. However under alternative (if null is false) as sample size increases one could always reject null. This result shows properties of simple statistical test. Also martingale is used in survival analysis to show properties of statistical procedures.