Characteristic function (probability theory)

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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:

[\varphi_X(t) = \operatorname\left(e^\right)\,]

where t is a real number and E denotes the expected value.

If F_X is the cumulative distribution function, then the characteristic function is given by the Riemann-Stieltjes integral

[\operatorname\left(e^\right) = \int_\Omega e^\,dF_X(x).\,]

In cases in which there is a probability density function, f_X, this becomes

[\operatorname\left(e^\right) = \int_^ e^ f_X(x)\,dx.]

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

Every probability distribution on R or on Rⁿ has a characteristic function, because one is integrating a bounded function over a space whose measure is finite.

Contents

1 The inversion theorem
2 The continuity theorem
3 Uses of characteristic functions
4 Related concepts

The inversion theorem

More than that, there is a bijection between cumulative probability distribution functions and characteristic functions. In other words, two distinct probability distributions never share the same characteristic function.

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

[F_X(y) - F_X(x) = \lim_ \frac \int_^ \frac - e^} \, \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.

The continuity theorem

If the sequence of characteristic functions of distributions F_n converges to the characteristic function of a distribution F, then F_n(x) converges to F(x) at every value of x at which F is continuous.

Uses of characteristic functions

Characteristic functions are particularly useful for dealing with functions of independent random variables. For example, if X₁, X₂, ..., X_n is a sequence of independent (and not necessarily identically distributed) random variables, and

[S_n = \sum_^n a_i X_i,]

where the a_i are constants, then the characteristic function for S_n is given by

[\varphi_(t)=\varphi_(a_1t)\varphi_(a_2t)\cdots \varphi_(a_nt).]

Because of the continuity theorem, characteristic functions are used in the most frequently seen proof of the central limit theorem.

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

[\operatorname\left(X^n\right) = (-i)^n\, \varphi_X^(0) = (-i)^n\, \left[frac varphi_X(t)right]_. ]

Related concepts

Related concepts include the moment-generating function and the probability-generating function. The characteristic function exists for all the probability distributions. However this is not the case for moment 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. In fact, the density function is equal to the Fourier transform of the characteristic function (up to a constant of proportionality and assuming the integral is defined)

[f_X(x) = \frac \int_^ \varphi_X(t)\,e^\, dt.]

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