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In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences. Suppose is a martingale (or super-martingale) and : almost surely. Then for all positive integers ''N'' and all positive reals ''t'', : And symmetrically (when ''X''''k'' is a sub-martingale): : If ''X'' is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound: : Azuma's inequality applied to the Doob martingale gives the method of bounded differences (MOBD) which is common in the analysis of randomized algorithms. ==Simple example of Azuma's inequality for coin flips== Let ''F''''i'' be a sequence of independent and identically distributed random coin flips (i.e., let ''F''''i'' be equally likely to be -1 or 1 independent of the other values of ''F''''i''). Defining yields a martingale with |''X''''k'' − ''X''''k''−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get : For example, if we set ''t'' proportional to ''N'', then this tells us that although the ''maximum'' possible value of ''X''''N'' scales linearly with ''N'', the ''probability'' that the sum scales linearly with ''N'' decreases exponentially fast with ''N''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Azuma's inequality」の詳細全文を読む スポンサード リンク
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