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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 Large deviations of Gaussian random functions ： ウィキペディア英語版 Large deviations of Gaussian random functions A random function – of either one variable (a random process), or two or more variables (a random field) – is called Gaussian if every finite-dimensional distribution is a multivariate normal distribution. Gaussian random fields on the sphere are useful (for example) when analysing * the anomalies in the cosmic microwave background radiation (see,〔Robert J. Adler, "On excursion sets, tube formulas and maxima of random fields", (The Annals of Applied Probability 2000, Vol. 10, No. 1, 1–74 ). (Special invited paper.)〕 pp. 8–9); * brain images obtained by positron emission tomography (see,〔 pp. 9–10). Sometimes, a value of a Gaussian random function deviates from its expected value by several standard deviations. This is a large deviation. Though rare in a small domain (of space or/and time), large deviations may be quite usual in a large domain. == Basic statement == Let $M$ be the maximal value of a Gaussian random function $X$ on the (two-dimensional) sphere. Assume that the expected value of $X$ is $0$ (at every point of the sphere), and the standard deviation of $X$ is $1$ (at every point of the sphere). Then, for large $a>0$, $P(M>a)$ is close to $C\; a\; \backslash exp(-a^2/2)\; +\; 2P(\backslash xi>a)$, where $\backslash xi$ is distributed $N(0,1)$ (the standard normal distribution), and $C$ is a constant; it does not depend on $a$, but depends on the correlation function of $X$ (see below). The relative error of the approximation decays exponentially for large $a$. The constant $C$ is easy to determine in the important special case described in terms of the directional derivative of $X$ at a given point (of the sphere) in a given direction (tangential to the sphere). The derivative is random, with zero expectation and some standard deviation. The latter may depend on the point and the direction. However, if it does not depend, then it is equal to $(\backslash pi/2)^\; C^$ (for the sphere of radius $1$). The coefficient $2$ before $P(\backslash xi>a)$ is in fact the Euler characteristic of the sphere (for the torus it vanishes). It is assumed that $X$ is twice continuously differentiable (almost surely), and reaches its maximum at a single point (almost surely). 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「Large deviations of Gaussian random functions」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.