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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 Mixing (mathematics) ： ウィキペディア英語版 Mixing (mathematics) In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, ''etc''. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity). ==Mixing in stochastic processes== Let $\backslash langle\; X\_t\; \backslash rangle\; =\; \backslash ,\; \backslash ldots\; \backslash \}$ be a sequence of random variables. Such a sequence is naturally endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a sigma algebra, the Borel sigma algebra; it is the smallest (coarsest) sigma algebra that contains the topology. Define a function $\backslash alpha(s)$, called the strong mixing coefficient, as :$\backslash alpha(s)\; \backslash equiv\; \backslash sup\; \backslash left\backslash ,\; B\backslash in\; X\_^\backslash infty\; \backslash ,\backslash right\backslash \}.$ In this definition, ''P'' is the probability measure on the sigma algebra. The symbol $X\_a^b$, with $-\backslash infty\; \backslash leq\; a\; \backslash leq\; b\; \backslash leq\; \backslash infty$ denotes a subalgebra of the sigma algebra; it is the set of cylinder sets that are specified between times ''a'' and ''b''. Given specific, fixed values $X\_a$, $X\_$, ''etc.'', of the random variable, at times $a$, $a+1$, ''etc.'', then it may be thought of as the sigma-algebra generated by :$\backslash .$ The process $\backslash langle\; X\_t\; \backslash rangle$ is strong mixing if $\backslash alpha(s)\backslash rightarrow\; 0$ as $s\backslash rightarrow\; \backslash infty$. One way to describe this is that strong mixing implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is independent. 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「Mixing (mathematics)」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.