Doob's Martingale convergence theorem

Doob's Martingale Convergence Theorem states that if you have a fair "game" or process where your expected future value equals your current value, and this process doesn't become overly extreme (bounded in average), then as time goes on, the process will tend to settle into a specific value—converging to a limit. In essence, under these conditions, a martingale stabilizes over the long run, ensuring predictable outcomes in the limit, despite the randomness at each step. This theorem is foundational in probability theory, especially in understanding how certain stochastic processes behave over time.