characteristic function

Let X be a random variableMathworldPlanetmath. The characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath of X is a function φX: defined by


that is, φX(t) is the expectation of the random variable eitX.

Given a random vector X¯=(X1,,Xn), the characteristic function of X¯, also called joint characteristic function of X1,,Xn, is a function φX¯:n defined by


where t¯=(t1,,tn) and t¯X¯=t1X1++tnXn (the dot product.)

Remark. If FX is the distribution functionMathworldPlanetmath associated to X, by the properties of expectation we have


which is known as the Fourier-Stieltjes transform of FX, and provides an alternate definition of the characteristic function. From this, it is clear that the characteristic function depends only on the distribution function of X, hence one can define the characteristic function associated to a distributionPlanetmathPlanetmath even when there is no random variable involved. This implies that two random variables with the same distribution must have the same characteristic function. It is also true that each characteristic function determines a unique distribution; hence the , since it characterizes the distribution function (see property 6.)


  1. 1.

    The characteristic function is bounded by 1, i.e. |φX(t)|1 for all t;

  2. 2.


  3. 3.

    φX(t)¯=φX(-t), where z¯ denotes the complex conjugate of z;

  4. 4.

    φX is uniformly continuous in ;

  5. 5.

    If X and Y are independent random variables, then φX+Y=φXφY;

  6. 6.

    The characteristic function determines the distribution function; hence, φX=φY if and only if FX=FY. This is a consequence of the inversion : Given a random variable X with characteristic function φ and distribution function F, if x and y are continuity points of F such that x<y, then

  7. 7.

    A random variable X has a symmetrical distribution (i.e. one such that FX=F-X) if and only if φX(t) for all t;

  8. 8.

    For real numbers a,b, φaX+b(t)=eitbφX(at);

  9. 9.

    If E|X|n<, then φX has continuousMathworldPlanetmathPlanetmath n-th derivativesPlanetmathPlanetmath and

    dkφXdtk(t)=φX(k)(t)=(ix)keitx𝑑FX(x),  1kn.

    Particularly, φX(k)(0)=ikEXk; characteristic functions are similar to moment generating functions in this sense.

Similar properties hold for joint characteristic functions. Other important result related to characteristic functions is the Paul Lévy continuity theorem.

Title characteristic function
Canonical name CharacteristicFunction1
Date of creation 2013-03-22 13:14:28
Last modified on 2013-03-22 13:14:28
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Definition
Classification msc 60E10
Synonym joint characteristic function
Related topic MomentGeneratingFunction
Related topic CumulantGeneratingFunction