Theorem of martingales

The theorem of martingales describes a situation where, in a fair game or process, the best prediction of the future value is always the current value—meaning there’s no predictable trend up or down. Mathematically, a martingale is a sequence of values where the expected next value equals the current one, given all past information. This property ensures that, on average, you cannot win or lose money over time if the game is fair. The theorem formalizes this idea, highlighting that no strategy can reliably outperform the fairness or randomness built into such processes.