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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 Free independence ： ウィキペディア英語版 Free independence In the mathematical theory of free probability, the notion of free independence was introduced by Dan Voiculescu.〔D. Voiculescu, K. Dykema, A. Nica, "Free Random Variables", CIRM Monograph Series, AMS, Providence, RI, 1992〕 The definition of free independence is parallel to the classical definition of independence, except that the role of Cartesian products of measure spaces (corresponding to tensor products of their function algebras) is played by the notion of a free product of (non-commutative) probability spaces. In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability analogs: the same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion of independence is replaced by free independence. Examples of this include: the free central limit theorem; notions of free convolution; existence of free stochastic calculus and so on. Let $(A,\backslash phi)$ be a non-commutative probability space, i.e. a unital algebra $A$ over $\backslash mathbb$ equipped with a unital linear functional $\backslash phi:A\backslash to\backslash mathbb$. As an example, one could take, for a probability measure $\backslash mu$, : $A\; =\; L^\backslash infty(\backslash mathbb,\backslash mu),\backslash phi(f)\; =\; \backslash int\; f(t)\backslash ,d\backslash mu(t).$ Another example may be $A=M\_N$, the algebra of $N\backslash times\; N$ matrices with the functional given by the normalized trace $\backslash phi=\backslash fracTr$. Even more generally, $A$ could be a von Neumann algebra and $\backslash phi$ a state on $A$. A final example is the group algebra $A=\backslash mathbb\backslash Gamma$ of a (discrete) group $\backslash Gamma$ with the functional $\backslash phi$ given by the group trace $\backslash phi\; (g)\; =\; \backslash delta\_,g\backslash in\; \backslash Gamma$. Let $\backslash$ be a family of unital subalgebras of $A$. Definition. The family $\backslash$ is called ''freely independent'' if $\backslash phi(x\_1\; x\_2\; \backslash cdots\; x\_n)\; =0$ whenever $\backslash phi(x\_j)=0$, $x\_j\; \backslash in\; A\_$ and $i(1)\backslash neq\; i(2),\; i(2)\backslash neq\; i(3),\backslash dots$. If $X\_i\backslash in\; A$, $i\backslash in\; I$ is a family of elements of $A$ (these can be thought of as random variables in $A$), they are called ''freely independent'' if the algebras $A\_i$ generated by $1$ and $X\_i$ are freely independent. == Examples of free independence == * Let $\backslash Gamma$ be the free product of groups $\backslash Gamma\_i,i\backslash in\; I$, let $A=\backslash mathbb\backslash Gamma$ be the group algebra, $\backslash phi(g)=\backslash delta\_$ be the group trace, and set $A\_i=\backslash mathbb\backslash Gamma\_i\backslash subset\; A$. Then $A\_i:i\backslash in\; I$ are freely independent. * Let $U\_i(N),i=1,2$ be $N\backslash times\; N$ unitary random matrices, taken independently at random from the $N\backslash times\; N$ unitary group (with respect to the Haar measure). Then $U\_1(N),U\_2(N)$ become asymptotically freely independent as $N\backslash to\backslash infty$. (Asymptotic freeness means that the definition of freeness holds in the limit as $N\backslash to\backslash infty$). * More generally, independent random matrices tend to be asymptotically freely independent, under certain conditions. 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「Free independence」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.