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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bernstein inequalities in probability theory ： ウィキペディア英語版 Bernstein inequalities (probability theory) In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let ''X''1, ..., ''X''''n'' be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive $\backslash varepsilon$, :$\backslash mathbf\; \backslash left\; (\backslash left|\backslash frac\backslash sum\_^n\; X\_i\backslash right|\; >\; \backslash varepsilon\; \backslash right\; )\; \backslash leq\; 2\backslash exp\; \backslash left\; (-\backslash frac)\}\; \backslash right).$ Bernstein inequalities were proved and published by Sergei Bernstein in the 1920s and 1930s.〔S.N.Bernstein, "On a modification of Chebyshev’s inequality and of the error formula of Laplace" vol. 4, #5 (original publication: Ann. Sci. Inst. Sav. Ukraine, Sect. Math. 1, 1924)〕〔S.N.Bernstein, "Theory of Probability" (Russian), Moscow, 1927〕〔J.V.Uspensky, "Introduction to Mathematical Probability", McGraw-Hill Book Company, 1937〕 Later, these inequalities were rediscovered several times in various forms. Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality. ==Some of the inequalities== 1. Let ''X''1, ..., ''X''''n'' be independent zero-mean random variables. Suppose that |''X'' ''i''| ≤ ''M'' almost surely, for all ''i''. Then, for all positive ''t'', :$\backslash mathbf\; \backslash left\; (\backslash sum\_^n\; X\_i\; >\; t\; \backslash right\; )\; \backslash leq\; \backslash exp\; \backslash left\; (\; -\backslash frac\; t^2\}\; Mt\}\; \backslash right\; ).$ 2. Let ''X''1, ..., ''X''''n'' be independent random variables. Suppose that for some positive real L and every integer ''k'' > 1, :$\backslash mathbf\; \backslash left\; ()\; \backslash leq\; \backslash tfrac\; \backslash mathbf\; \backslash left()\; L^\; k!$ Then : 3. Let ''X''1, ..., ''X''''n'' be independent random variables. Suppose that :$\backslash mathbf\; \backslash left()\; \backslash leq\; \backslash frac\; \backslash left(\backslash frac\backslash right)^$ for all integer ''k'' > 3. Denote :$A\_k\; =\; \backslash sum\; \backslash mathbf\; \backslash left\; (X\_i^k\backslash right).$ Then, : 4. Bernstein also proved generalizations of the inequalities above to weakly dependent random variables. For example, inequality (2) can be extended as follows. Let ''X''1, ..., ''X''''n'' be possibly non-independent random variables. Suppose that for all integer ''i'' > 0, :$\backslash begin$ \mathbf \left (X_i | X_1, \dots, X_ \right ) &= 0, \\ \mathbf \left (X_i^2 | X_1, \dots, X_ \right ) &\leq R_i \mathbf \left (X_i^2 \right ), \\ \mathbf \left (X_i^k | X_1, \dots, X_ \right ) &\leq \tfrac \mathbf \left(X_i^2 | X_1, \dots, X_ \right ) L^ k! \end Then : More general results for martingales can be found in Fan et al. (2012). 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「Bernstein inequalities (probability theory)」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.