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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 Asymptotic theory ： ウィキペディア英語版 Asymptotic theory Asymptotic theory or large-sample theory is the branch of mathematics which studies asymptotic expansions. An example of an asymptotic result is the prime number theorem: Let π(''x'') be the number of prime numbers that are smaller than or equal to ''x''. Then the limit :$\backslash lim\_\backslash frac$ exists, and it is equal to 1. Asymptotic theory ("asymptotics") is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory. == Asymptotic distribution == In mathematics and statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables :$Z\_i$ for $i\; =\; 1$ to $n$ for some positive integer $n$. An asymptotic distribution allows $i$ to range without bound, that is, $n$ is infinite. A special case of an asymptotic distribution is when the late entries go to zero—that is, the Zi go to 0 as i goes to infinity. Some instances of "asymptotic distribution" refer only to this special case. This is based on the notion of an asymptotic function which cleanly approaches a constant value (the ''asymptote'') as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon. An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation :$y\; =\; \backslash frac$, $y$ becomes arbitrarily small in magnitude as $x$ increases. It is often used in time series analysis. In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. If φ''n'' is a sequence of continuous functions on some domain, and if ''L'' is a (possibly infinite) limit point of the domain, then the sequence constitutes an asymptotic scale if for every ''n'', $\backslash varphi\_(x)\; =\; o(\backslash varphi\_n(x))\; \backslash \; (x\; \backslash rightarrow\; L)$. If ''f'' is a continuous function on the domain of the asymptotic scale, then an asymptotic expansion of ''f'' with respect to the scale is a formal series $\backslash sum\_^\backslash infty\; a\_n\; \backslash varphi\_(x)$ such that, for any fixed ''N'', :$f(x)\; =\; \backslash sum\_^N\; a\_n\; \backslash varphi\_(x)\; +\; O(\backslash varphi\_(x))\; \backslash \; (x\; \backslash rightarrow\; L).$ In this case, we write :$f(x)\; \backslash sim\; \backslash sum\_^\backslash infty\; a\_n\; \backslash varphi\_n(x)\; \backslash \; (x\; \backslash rightarrow\; L)$. See asymptotic analysis and big O notation for the notation. The most common type of asymptotic expansion is a power series in either positive or negative terms. While a convergent Taylor series fits the definition as given, a non-convergent series is what is usually intended by the phrase. Methods of generating such expansions include the Euler–Maclaurin formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. 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「Asymptotic theory」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.