Asymptotic theory
Encyclopedia
Asymptotic theory or large sample 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 number
s that are smaller than or equal to x.
The limit
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.
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 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 φn 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,
. 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
such that, for any fixed N,
In this case, we write
.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
which studies properties of asymptotic expansions.
The most known result of this field is the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
:
Let π(x) be the number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s that are smaller than or equal to x.
The limit
exists, and is equal to 1.
Some results often neglected include the probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of the likelihood ratio statistic
Statistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...
and the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the deviance in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, results that are used daily by applied statisticians.
Asymptotic distribution
In mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, an asymptotic distribution
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...
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
- Zi
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
Asymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
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
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
analysis.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
) 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 φn 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,
. If f is a continuous function on the domain of the asymptotic scale, then an asymptotic expansion off with respect to the scale is a formal series
such that, for any fixed N,In this case, we write
.See asymptotic analysis
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
and big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
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
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
transforms. Repeated integration by parts will often lead to an asymptotic expansion.
Examples of asymptotic expansions
- Gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
-
- Exponential integralExponential integralIn mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...
-
>where
are Bernoulli numbers and 
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 
.
- Error functionError functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
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
The expression on the left is valid on the entire complex plane
, while the right hand side converges only for 
. Multiplying by 
and integrating both sides yields
The integral on the left hand side can be expressed in terms of the exponential integralExponential integralIn mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...
. The integral on the right hand side, after the substitution
, may be recognized as the gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. Evaluating both, one obtains the asymptotic expansion
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
. Substituting 
and noting that 
results in the asymptotic expansion given earlier in this article.
External links
- Error function
- Exponential integral
