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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 convolution ： ウィキペディア英語版 Free convolution Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of random variables). These operations have some interpretations in terms of empirical spectral measures of random matrices.〔Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.〕 The notion of free convolution was introduced by Voiculescu.〔Voiculescu, D., Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323–346〕〔Voiculescu, D., Multiplication of certain noncommuting random variables , J. Operator Theory 18 (1987), 2223–2235〕 == Free additive convolution == Let $\backslash mu$ and $\backslash nu$ be two probability measures on the real line, and assume that $X$ is a random variable in a non commutative probability space with law $\backslash mu$ and $Y$ is a random variable in the same non commutative probability space with law $\backslash nu$. Assume finally that $X$ and $Y$ are freely independent. Then the free additive convolution $\backslash mu\backslash boxplus\backslash nu$ is the law of $X+Y$. Random matrices interpretation: if $A$ and $B$ are some independent $n$ by $n$ Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the empirical spectral measures of $A$ and $B$ tend respectively to $\backslash mu$ and $\backslash nu$ as $n$ tends to infinity, then the empirical spectral measure of $A+B$ tends to $\backslash mu\backslash boxplus\backslash nu$.〔Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.〕 In many cases, it is possible to compute the probability measure $\backslash mu\backslash boxplus\backslash nu$ explicitly by using complex-analytic techniques and the R-transform of the measures $\backslash mu$ and $\backslash nu$. 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 convolution」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.