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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 Bernoulli shift ： ウィキペディア英語版 Bernoulli scheme In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.〔P. Shields, ''The theory of Bernoulli shifts'', Univ. Chicago Press (1973)〕〔Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in ''Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces'', Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 0-19-853390-X〕 Bernoulli schemes are important in the study of dynamical systems, as most such systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift.〔Pierre Gaspard, ''Chaos, scattering and statistical mechanics''(1998), Cambridge University press〕 This is essentially the Markov partition. The term ''shift'' is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal. ==Definition== A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of ''N'' distinct possible values, with the outcome ''i'' occurring with probability $p\_i$, with ''i'' = 1, ..., ''N'', and :$\backslash sum\_^N\; p\_i\; =\; 1.$ The sample space is usually denoted as :$X=\backslash ^\backslash mathbb$ as a shorthand for :$X=\backslash \; \backslash ;\; \backslash forall\; k\; \backslash in\; \backslash mathbb\; \backslash \}.$ The associated measure is called the Bernoulli measure :$\backslash mu\; =\; \backslash ^\backslash mathbb$ The σ-algebra $\backslash mathcal$ on ''X'' is the product sigma algebra; that is, it is the (countable) direct product of the σ-algebras of the finite set . Thus, the triplet :$(X,\backslash mathcal,\backslash mu)$ is a measure space. A basis of $\backslash mathcal$ is the cylinder sets. Given a cylinder set $(x\_1,\backslash ldots,x\_n)$, its measure is :$\backslash mu\backslash left((x\_1,\backslash ldots,x\_n)\backslash right)=$ \prod_^n p_ The equivalent expression, using the notation of probability theory, is :$\backslash mu\backslash left((x\_1,\backslash ldots,x\_n)\backslash right)=$ \mathrm(X_0=x_0, X_1=x_1, \ldots, X_n=x_n) for the random variables $\backslash$ The Bernoulli scheme, as any stochastic process, may be viewed as a dynamical system by endowing it with the shift operator ''T'' where :$Tx\_k\; =\; x\_.$ Since the outcomes are independent, the shift preserves the measure, and thus ''T'' is a measure-preserving transformation. The quadruplet :$(X,\backslash mathcal,\backslash mu,\; T)$ is a measure-preserving dynamical system, and is called a Bernoulli scheme or a Bernoulli shift. It is often denoted by :$BS(p)=BS(p\_1,\backslash ldots,p\_N).$ The ''N'' = 2 Bernoulli scheme is called a Bernoulli process. The Bernoulli shift can be understood as a special case of the Markov shift, where all entries in the adjacency matrix are one, the corresponding graph thus being a clique. 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「Bernoulli scheme」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.