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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 Marchenko–Pastur distribution ： ウィキペディア英語版 Marchenko–Pastur distribution In random matrix theory, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices. The theorem is named after Ukrainian mathematicians Vladimir Marchenko and Leonid Pastur who proved this result in 1967. If $X$ denotes a $M\backslash times\; N$ random matrix whose entries are independent identically distributed random variables with mean 0 and variance , let : $Y\_N\; =\; N^\; X\; X^T\; \backslash ,$ and let $\backslash lambda\_1,\backslash ,\; \backslash lambda\_2,\; \backslash ,\backslash dots,\backslash ,\; \backslash lambda\_M$ be the eigenvalues of $Y\_N$ (viewed as random variables). Finally, consider the random measure : $\backslash mu\_M\; (A)\; =\; \backslash frac\; \backslash \#\; \backslash left\backslash ,\; \backslash quad\; A\; \backslash subset\; \backslash mathbb.$ Theorem. Assume that $M,\backslash ,N\; \backslash ,\backslash to\backslash ,\; \backslash infty$ so that the ratio $M/N\; \backslash ,\backslash to\backslash ,\; \backslash lambda\; \backslash in\; (0,\; +\backslash infty)$. Then $\backslash mu\_\; \backslash ,\backslash to\backslash ,\; \backslash mu$ (in weak * topology in distribution), where : \nu(A),& \text 0\leq \lambda \leq 1, \end and : $d\backslash nu(x)\; =\; \backslash frac\; \backslash frac)\}\}\; \backslash ,\backslash mathbf\_,">\backslash lambda\_\; )\}\backslash ,\; dx$ with : $\backslash lambda\_\; =\; \backslash sigma^2(1\; \backslash pm\; \backslash sqrt)^2.\; \backslash ,$ The Marchenko–Pastur law also arises as the free Poisson law in free probability theory, having rate $1/\backslash lambda$ and jump size $\backslash sigma^2$. ==See also == * Wigner semicircle distribution 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「Marchenko–Pastur distribution」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.