Azuma's inequality is a result in probability theory that provides a concentration bound for submartingales (or sometimes martingales) given specific conditions. More specifically, it can be used to show how tightly concentrated the values of a martingale are around its expected value.
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If a martingale has increments with almost surely for deterministic , then each one-sided deviation of magnitude from has probability at most . If all vanish, the martingale is constant. The inequality gives concentration from bounded increments without requiring independent increments.