Doob's Martingale Theory

Doob's Martingale Theory is a concept from probability theory that deals with fair games. In simple terms, it describes a sequence of random variables (like betting outcomes) where the future expected value, given all past information, is equal to the current value. This means that, on average, you can't predict a gain or loss based on previous results; the game is "fair." It has important applications in finance, gambling, and statistics, as it helps analyze systems where outcomes are uncertain but structured, guiding decisions based on probabilistic reasoning rather than deterministic forecasts.