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单词 CharacteristicFunction1
释义

characteristic function


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

φX(t)=EeitX=Ecos(tX)+iEsin(tX),

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

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

φX¯(t)=Eeit¯X¯,

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

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

φX(t)=eitx𝑑FX(x),

which is known as the Fourier-Stieltjes transform of FX, and provides an alternatedefinition of the characteristic function. From this, it is clear that the characteristicfunction depends only on the distribution function of X, hence one can define the characteristicfunction 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 samecharacteristic function. It is also true that each characteristic function determinesa unique distribution; hence the , since it characterizes the distribution function (see property 6.)

Properties

  1. 1.

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

  2. 2.

    φX(0)=1;

  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 Xwith characteristic function φ and distribution function F, if x and yare continuity points of F such that x<y, then

    F(x)-F(y)=12πlims-sse-itx-e-ityitφ(t)𝑑t;
  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 tomoment generating functions in this sense.

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

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更新时间:2025/5/4 1:47:26