Doob's martingale inequality

Doob's Martingale Inequality is a fundamental result in the theory of martingales, which are stochastic processes that model fair game scenarios. Specifically, Doob's inequality provides bounds on the probabilities related to the maximum of a martingale. There are a couple of versions of Doob's Martingale Inequality, but the most common one deals with a bounded integrable martingale.