 Azuma's inequality

If a martingale has increments with $|M_i-M_{i-1}|\le c_i$ almost surely for deterministic $c_i$, then each one-sided deviation of magnitude $t>0$ from $M_0$ has probability at most $\exp (-t^2/(2{\sum }_i{c}_i^2))$. If all $c_i$ vanish, the martingale is constant. The inequality gives concentration from bounded increments without requiring independent increments.

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.