Doob's Optional Stopping Theorem

Doob's Optional Stopping Theorem is a principle in probability theory, particularly in the study of martingales, which are sequences of random variables that represent fair games. The theorem states that under certain conditions, if you stop observing this sequence at a random time, the expected value of the amount you have when you stop is the same as it was at the beginning. Essentially, it illustrates that, when playing fair games, stopping at a random point won’t change your expected outcome, provided that specific technical conditions are met regarding the stopping time and the randomness of the sequence.