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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 Lindeberg's condition ： ウィキペディア英語版 Lindeberg's condition In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables. Unlike the classical CLT, which requires that the random variables in question have finite mean and variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite mean and variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. ==Statement== Let $(\backslash Omega,\; \backslash mathcal,\; \backslash mathbb)$ be a probability space, and $X\_k\; :\; \backslash Omega\; \backslash to\; \backslash mathbb,\backslash ,\backslash ,\; k\; \backslash in\; \backslash mathbb$, be ''independent'' random variables defined on that space. Assume the expected values $\backslash mathbb\backslash ,()\; =\; \backslash mu\_k$ and variances $\backslash mathrm\backslash ,()\; =\; \backslash sigma\_k^2$ exist and are finite. Also let $s\_n^2\; :=\; \backslash sum\_^n\; \backslash sigma\_k^2\; .$ If this sequence of independent random variables $X\_k$ satisfies Lindeberg's condition: :$\backslash lim\_\; \backslash frac\backslash sum\_^\; \backslash mathbb\backslash big(-\; \backslash mu\_k)^2\; \backslash cdot\; \backslash mathbf\_\; is\; theindicator\; function,\; then\; thecentral\; limit\; theoremholds,\; i.e.\; the\; random\; variables$ :$Z\_n\; :=\; \backslash frac$ converge in distribution to a standard normal random variable as $n\; \backslash to\; \backslash infty.$ Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general). However, if the sequence of independent random variables in question satisfies :$\backslash max\_\; \backslash frac\; \backslash to\; 0,\; \backslash quad\; \backslash text\; n\; \backslash to\; \backslash infty,$ then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds. 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「Lindeberg's condition」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.