Characteristic function

Some mathematicians use the phrase characteristic function synonymously with "indicator function". The indicator function of a subset A of a set B is the function with ___domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A.

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In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:

${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right)=\int _{\Omega }e^{itx}\,dF_{X}(x)=\int _{-\infty }^{\infty }f_{X}(x)\,e^{itx}\,dx.}$

Here t is a real number, E denotes the expected value and F is the cumulative distribution function. The last form is valid only when f--the probability density function--exists. The form preceding it is a Riemann-Stieltjes integral and is valid regardless of whether a density function exists.

If X is a vector-valued random variable, one takes the argument t to be a vector and tX to be a dot product.

Characteristic function exists for any random variable. More than that, there is a bijection between cumulative probability functions and characteristic functions. In other words, each cumulative probability function has one and only one characteristic function that corresponds to it.

Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function F:

${\displaystyle F_{X}(y)-F_{X}(x)=\lim _{\tau \to +\infty }{\frac {1}{2\pi }}\int _{-\tau }^{+\tau }{\frac {e^{-itx}-e^{-ity}}{it}}\,\varphi _{X}(t)\,dt.}$

In general this is an improper integral; the function being integrated may be only conditionally integrable rather than Lebesgue-integrable, i.e. the integral of its absolute value may be infinite.

Characteristic functions can also be used to find moments of random variable. Provided that n-th moment exists, characteristic function can be differentiated n times and

${\displaystyle \operatorname {E} \left(X^{n}\right)=i^{n}\,\varphi _{X}^{(n)}(0)=i^{n}\,\left.{\frac {d^{n}}{dt^{n}}}\right|_{t=0}\varphi _{X}(t)}$

Related concepts include the moment-generating function and the probability-generating function.

The characteristic function is closely related to the Fourier transform: the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.