翻訳と辞書 Words near each other ・ Azulejo ・ Azulene ・ Azules de Veracruz ・ Azulewicz coat of arms ・ Azuli Records ・ Azuloscurocasinegro ・ Azuma ・ Azuma (name) ・ Azuma Kagami ・ Azuma Konno ・ Azuma Koshiishi ・ Azuma Leasing ・ Azuma Morisaki ・ Azuma Sakamoto ・ Azuma Yano ・ Azuma's inequality ・ Azuma, Gunma (Agatsuma) ・ Azuma, Gunma (Sawa) ・ Azuma, Gunma (Seta) ・ Azuma, Gunma (Tone) ・ Azuma, Ibaraki ・ Azuma, Kagoshima ・ Azuma, Nagasaki ・ Azuma-gumi ・ Azumafuji Kin'ichi ・ Azumah ・ Azumah Nelson ・ Azumah Nelson Sports Complex ・ Azumaia ・ Azumanga Daioh Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Azuma's inequality ： ウィキペディア英語版 Azuma's inequality 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'', :$P(X\_N\; -\; X\_0\; \backslash geq\; t)\; \backslash leq\; \backslash exp\backslash left\; (\; \backslash right).$ And symmetrically (when ''X''''k'' is a sub-martingale): :$P(X\_N\; -\; X\_0\; \backslash leq\; -t)\; \backslash leq\; \backslash exp\backslash left\; (\; \backslash right).$ If ''X'' is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound: :$P(|X\_N\; -\; X\_0|\; \backslash geq\; t)\; \backslash leq\; 2\backslash exp\backslash left\; (\; \backslash right).$ 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 $X\_i\; =\; \backslash sum\_^i\; F\_j$ yields a martingale with |''X''''k'' − ''X''''k''−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get :$\backslash Pr(>\; t)\; \backslash leq\; \backslash exp\backslash left(\backslash frac\backslash right).$ 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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.