Doob's theorem

Doob's theorem relates to the behavior of certain types of random processes called "martingales," which model fair games where future expectations equal present values. The theorem states that if such a process remains bounded and exhibits a specific kind of regularity over time, it will stabilize and converge to a constant value. Essentially, under these conditions, the process's future predictions become predictable, settling into a steady state. This concept is fundamental in probability theory and helps in understanding long-term behaviors in financial models, statistical processes, and other areas involving uncertainty.