characteristic function

Also found in: Dictionary, Acronyms, Wikipedia.

characteristic function

[‚kar·ik·tə′ris·tik ′fəŋk·shən]
The function χA defined for any subset A of a set by setting χA (x) = 1 if x is in A and χA = 0 if x is not in A. Also known as indicator function.
A function, such as the point characteristic function or the principal function, which is the integral of some property of an optical or mechanical system over time or over the path followed by the system, and whose value for a path actually followed by a system is a maximum or a minimum with respect to nearby paths with the same end points.
A function that uniquely defines a probability distribution; it is equal to √(2π) times the Fourier transform of the frequency function of the distribution.

Characteristic Function


in mathematics:

(1) An eigenfunction.

(2) The characteristic, or indicator, function of a set A is a function f(x) that is defined on some set E containing A and that assumes the value f(x) = 1 if x is in A and the value f(x) = 0 if x is not in A.

(3) In probability theory, the characteristic function fx(t) of a random variable X is the mathematical expectation of the quantity exp (it X). For a random variable with probability density function PX(x), this definition yields the formula

For example, for a random variable having a normal distribution with parameters a and σ, the characteristic function is

The characteristic function has several noteworthy properties. To every random variable X there corresponds a definite characteristic function. The probability distribution for X is uniquely determined by fX(t). When independent random variables are added, the corresponding characteristic functions are multiplied. If the concept of closeness is suitably defined, to random variables with close distributions there correspond characteristic functions that differ little from each other, and, conversely, to close characteristic functions there correspond random variables with close distributions. These properties underlie the applications of characteristic functions, particularly applications to the derivation of the limit theorems of probability theory.

A mathematical apparatus more or less equivalent to that of the characteristic function was first used by P. Laplace in 1812, but the full power of the characteristic function method was demonstrated in 1911 by A. M. Liapunov, who used the method in obtaining the theorem that bears his name.

The concept of characteristic function can be generalized to finite and infinite systems of random variables—that is, to random vectors and stochastic processes. The theory of characteristic functions has much in common with the theory of Fourier integrals.


Gnedenko, B. V. Kursteorii veroiatnostei, 5th ed. Moscow, 1969.
Prokhorov, Iu. V., and Iu. A. Rozanov. Teoriia veroiatnostei, 2nd ed. Moscow, 1973.

characteristic function

The characteristic function of set returns True if its argument is an element of the set and False otherwise.
References in periodicals archive ?
From the expression of characteristic function a conclusion can be made that applying the proposed method of strobo pulse formation and synchronization with the necessary vibration phase, the intensity of the reconstructed image described by this function depends on duration of the strobo light pulse and on the ratio of parameters of different vibration components.
However, note that when observations are not enough to estimate the stable characteristic function, it may produce a value of [?
perform a change of measure on the characteristic function in order to make the price process a martingale
It can be verified independently that [PHI](t) is the characteristic function of a variable whose density is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The reserve virtual surface will be submitting (block 5) these some procedures on which the real surface during measuring, so: during sampling, filtration, estimation the surface roughness parameters, and other characteristic function (Cieloszyk, 2004).
Two very important methods have been developed that solve this problem in different ways: (1) Fourier inversion of the characteristic function of F(x) and (2) the Panjer recursion formula.
It is well known that the characteristic function of a random vector always exists and there exists a one-to-one correspondence between distribution functions and characteristic functions.
The balancedness of the characteristic function in games with transferable utility (TU-games) determines, whether a grand coalition or restricted grand coalition is stable against the blocking by subcoalitions/subnetworks.
0]([lambda]) is called the characteristic function for the boundary value problems [L.
The three usual ways to define a jump diffusion process are by specifying its dynamics or by expressing its exponential argument, or by giving its characteristic function.
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The characteristic function normalized value of the company for market segment i at time h

Full browser ?