Asymptotic theory

Asymptotic theory

Asymptotic theory is the branch of mathematics which studies properties of asymptotic expansions.

The most known result of this field is the prime number theorem:Let π("x") be the number of prime numbers that are smaller than or equal to "x".The limit

:lim_{x ightarrowinfty}frac{pi(x)ln(x)}{x}

exists, and is equal to 1.

Some results often neglected include the probability distribution of the likelihood ratio statistic and the expected value of the deviance in statistics, results that are used daily by applied statisticians.

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


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" = 1/"x",

"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 sequenceconstitutes an asymptotic scale if for every "n",varphi_{n+1}(x) = o(varphi_n(x)) (x ightarrow 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 sum_{n=0}^infty a_n varphi_{n}(x) such that, for any fixed "N",:f(x) = sum_{n=0}^N a_n varphi_{n}(x) + O(varphi_{N+1}(x)) (x ightarrow L).In this case, we write: f(x) sim sum_{n=0}^infty a_n varphi_n(x) (x ightarrow L).See asymptotic analysis and big O notation for the notation.

The most common type of asymptotic expansion is a power series in either positiveor negative terms. While a convergent Taylor series fits the definition asgiven, a non-convergent series is what is usually intended by the phrase. Methods of generating such expansions include the Euler–Maclaurin formula andintegral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.

Examples of asymptotic expansions

* Gamma function

::frac{e^x}{x^x sqrt{2pi x Gamma(x+1) sim 1+frac{1}{12x}+frac{1}{288x^2}-frac{139}{51840x^3}-cdots (x ightarrow infty)

* Exponential integral

::xe^xE_1(x) sim sum_{n=0}^infty frac{(-1)^nn!}{x^n} (x ightarrow infty)

* Riemann zeta function

::zeta(s) sim sum_{n=1}^{N-1}n^{-s} + frac{N^{1-s{s-1} +N^{-s} sum_{m=1}^infty frac{B_{2m} s^{overline{2m-1}{(2m)! N^{2m-1
where B_{2m} are Bernoulli numbers and s^{overline{2m-1 is a rising factorial. This expansion is valid for all complex "s" and is often used to compute the zeta function by using a large enough value of "N", for instance N > |s|.

* Error function

:: sqrt{pi}x e^{x^2}{ m erfc}(x) = 1+sum_{n=1}^infty (-1)^n frac{(2n)!}{n!(2x)^{2n.

Detailed example

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. Thus, for example, one may start with the ordinary series

:frac{1}{1-w}=sum_{n=0}^infty w^n

The expression on the left is valid on the entire complex plane w e 1, while the right hand side converges only for |w|< 1. Multiplying by e^{-w/t} and integrating both sides yields

:int_0^infty frac{e^{-w/t{1-w} dw = sum_{n=0}^infty t^{n+1} int_0^infty e^{-u} u^n du

The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution u=w/t, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion

:e^{-1/t}; operatorname{Ei}left(frac{1}{t} ight) = sum _{n=0}^infty n! ; t^{n+1}

Here, the right hand side is clearly not convergent for any non-zero value of "t". However, by keeping "t" small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of operatorname{Ei}(1/t). Substituting x=-1/t and noting that operatorname{Ei}(x)=-E_1(-x) results in the asymptotic expansion given earlier in this article.


* Hardy, G. H., "Divergent Series", Oxford University Press, 1949
* Paris, R. B. and Kaminsky, D., "Asymptotics and Mellin-Barnes Integrals", Cambridge University Press, 2001
* Whittaker, E. T. and Watson, G. N., "A Course in Modern Analysis", fourth edition, Cambridge University Press, 1963

External links

* [ A paper on time series analysis using asymptotic distribution]
* []

Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Asymptotic analysis — This article is about the comparison of functions as inputs approach infinite. For asymptotes in geometry, see asymptotic curve. In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has… …   Wikipedia

  • Asymptotic freedom — In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i.e. length scales that asymptotically converge to zero (or …   Wikipedia

  • Asymptotic gain model — The asymptotic gain model [Middlebrook, RD: Design oriented analysis of feedback amplifiers ; Proc. of National Electronics Conference, Vol. XX, Oct. 1964, pp. 1 4] G {infin} T} . while in classical feedback theory, in terms of the open loop gain …   Wikipedia

  • Asymptotic computational complexity — In computational complexity theory, asymptotic computational complexity is the usage of the asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, commonly associated with the usage of the big …   Wikipedia

  • Asymptotic equipartition property — In information theory the asymptotic equipartition property (AEP) is a general property of the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories of compression.Roughly speaking, the theorem… …   Wikipedia

  • Theory of computation — In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata… …   Wikipedia

  • Asymptotic stability — See also Lyapunov stability for an alternate definition used in dynamical systems. In control theory, a continuous linear time invariant system is asymptotically stable if and only if the system has eigenvalues only with strictly negative real… …   Wikipedia

  • Contiguity (probability theory) — In probability theory, two sequences of probability measures are said to be contiguous if asymptotically they share the same support. Thus the notion of contiguity extends the concept of absolute continuity to the sequences of measures. The… …   Wikipedia

  • Geometric group theory — is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the… …   Wikipedia

  • Additive number theory — In mathematics, additive number theory is a branch of number theory that studies ways to express an integer as the sum of integers in a set. Two classical problem in this area of number theory are the Goldbach conjecture and Waring s problem.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”