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.