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

\lim_{x\rightarrow\infty}\frac{\pi(x)\ln(x)}{x}

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 Z_i 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 = \frac{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 sequence constitutes an asymptotic scale if for every n, $\varphi_{n+1}(x) = o(\varphi_n(x)) \ (x \rightarrow L)$ . 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 $\sum_{n=0}^\infty a_n \varphi_{n}(x)$ such that, for any fixed N,

f(x) = \sum_{n=0}^N a_n \varphi_{n}(x) + O(\varphi_{N+1}(x)) \ (x \rightarrow L).

In this case, we write

f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \ (x \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.

Examples of asymptotic expansions

Gamma function

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

Exponential integral

xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \rightarrow \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}{\rm 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 $w\ne 1$ , 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}\right) = \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.

References

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

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