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In probability theory Azuma's inequality gives a concentration result for the values of martingales that have bounded differences. Formally, it says that:
<math>P(X_N \geq X_0 + t) \leq \exp\left ({-t^2 \over 2 \sum_{k=1}^{N}c_k^2} \right) <math>
if <math>X_k<math> is a martingale, and if
<math>|X_k - X_{k-1}| < c_k<math>.
Azuma's inequality applied to the Doob martingale gives the method of bounded differences (MOBD) which is common in the analysis of
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