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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:## Lévy continuity theorem

The core of the Lévy continuity theorem states that a sequence of random variables $scriptstyle\; (X\_n)\_\{n=1\}^infty$ where each $scriptstyle\; X\_n$ has a characteristic function $scriptstyle\; varphi\_n$
will converge in distribution towards a random variable $scriptstyle\; X$,
## The inversion theorem

## Bochner-Khinchin theorem

An arbitrary function $scriptstyle\; varphi$ is a characteristic function corresponding to some probability law $scriptstyle\; mu$ if and only if the following three conditions are satisfied: ## Uses of characteristic functions

### Basic properties

Characteristic functions are particularly useful for dealing with functions of independent random variables. For example, if X_{1}, X_{2}, ..., X_{n} is a sequence of independent (and not necessarily identically distributed) random variables, and### Moments

Characteristic functions can also be used to find moments of a random variable. Provided that the n^{th} moment exists, characteristic function can be differentiated n times and### An example

The Gamma distribution with scale parameter θ and a shape parameter k has the characteristic function
_{1} + k_{2}, and we therefore conclude
## Multivariate characteristic functions

### Example

If $Xsim\; N(0,Sigma)$ is a multivariate Gaussian with zero mean, then
# int_{xin R^n}frac{1}{left|2piSigmaright|^{1/2}}e^{-frac{1}{2}x^TSigma^{-1}x}cdot e^{itcdot x}dx

e^{-frac{1}{2}t^TSigma t}.
## Matrix-valued random variables

## Related concepts

Related concepts include the moment-generating function and the probability-generating function. The characteristic function exists for all probability distributions. However this is not the case for moment generating function.## References

- $varphi\_X(t)\; =\; operatorname\{E\}left(e^\{itX\}right),$

where t is a real number, i is the imaginary unit, 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

- $operatorname\{E\}left(e^\{itX\}right)\; =\; int\_\{-infty\}^\{infty\}\; e^\{itx\},dF\_X(x).,$

In cases in which there is a probability density function, f_{X}, this becomes

- $operatorname\{E\}left(e^\{itX\}right)\; =\; int\_\{-infty\}^\{infty\}\; e^\{itx\}\; f\_X(x),dx.$

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 R^{n} has a characteristic function, because one is integrating a bounded function over a space whose measure is finite, and for every characteristic function there is exactly one probability distribution.

The characteristic function of a symmetric PDF (that is, one with $p(x)=p(-x)$) is real, because the imaginary components obtained from $x>0$ cancel those from $x<0$.

- $X\_n\; xrightarrow\{mathcal\; D\}\; X\; qquadtextrm\{as\}qquad\; n\; to\; infty$

- $varphi\_n\; quad\; xrightarrow\{textrm\{pointwise\}\}\; quad\; varphi\; qquadtextrm\{as\}qquad\; n\; to\; infty$

The Lévy continuity theorem can be used to prove the weak law of large numbers, see the proof using convergence of characteristic functions.

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:

- $F\_X(y)\; -\; F\_X(x)\; =\; lim\_\{tau\; to\; +infty\}\; frac\{1\}\; \{2pi\}$

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.

Reference: see (P. Levy, Calcul des probabilites, Gauthier-Villars, Paris, 1925. p166)

(1) $scriptstyle\; varphi\; ,$ is continuous

(2) $scriptstyle\; varphi(0)\; =\; 1\; ,$

(3) $scriptstyle\; varphi\; ,$ is a positive definite function (note that this is a complicated condition which is not equivalent to $scriptstyle\; varphi\; >0$).

Because of the continuity theorem, characteristic functions are used in the most frequently seen proof of the central limit theorem. The main trick involved in making calculations with a characteristic function is recognizing the function as the characteristic function of a particular distribution.

- $S\_n\; =\; sum\_\{i=1\}^n\; a\_i\; X\_i,,!$

where the a_{i} are constants, then the characteristic function for S_{n} is given by

- $$

In particular, $varphi\_\{X+Y\}(t)\; =\; varphi\_X(t)varphi\_Y(t)$. To see this, write out the definition of characteristic function:

- $varphi\_\{X+Y\}(t)=Eleft(e^\{it(X+Y)\}right)=Eleft(e^\{itX\}e^\{itY\}right)=Eleft(e^\{itX\}right)Eleft(e^\{itY\}right)=varphi\_X(t)\; varphi\_Y(t)$.

Observe that the independence of $X$ and $Y$ is required to establish the equality of the third and fourth expressions.

Another special case of interest is when $a\_i=1/n$ and then $S\_n$ is the sample mean. In this case, writing $overline\{X\}$ for the mean,

- $varphi\_\{overline\{X\}\}(t)=left(varphi\_X(t/n)right)^n.$

- $operatorname\{E\}left(X^nright)\; =\; i^\{-n\},\; varphi\_X^\{(n)\}(0)$

For example, suppose $X$ has a standard Cauchy distribution. Then $varphi\_X(t)=e^\{-|t|\}$. See how this is not differentiable at $t=0$, showing that the Cauchy distribution has no expectation. Also see that the characteristic function of the sample mean $overline\{X\}$ of $n$ independent observations has characteristic function $varphi\_\{overline\{X\}\}(t)=(e^\{-|t|/n\})^n=e^\{-|t|\}$, using the result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself.

The logarithm of a characteristic function is a cumulant generating function, which is useful for finding cumulants.

- $(1\; -\; theta,i,t)^\{-k\}.$

- $X\; ~sim\; Gamma(k\_1,theta)\; mbox\{\; and\; \}\; Y\; sim\; Gamma(k\_2,theta)$

- $varphi\_X(t)=(1\; -\; theta,i,t)^\{-k\_1\},,qquad\; varphi\_Y(t)=(1\; -\; theta,i,t)^\{-k\_2\}$

- $varphi\_\{X+Y\}(t)=varphi\_X(t)varphi\_Y(t)=(1\; -\; theta,i,t)^\{-k\_1\}(1\; -\; theta,i,t)^\{-k\_2\}=left(1\; -\; theta,i,tright)^\{-(k\_1+k\_2)\}.$

- $X+Y\; sim\; Gamma(k\_1+k\_2,theta)\; ,$

- $forall\; i\; in\; \{1,ldots,\; n\}\; :\; X\_i\; sim\; Gamma(k\_i,theta)\; qquad\; Rightarrow\; qquad\; sum\_\{i=1\}^n\; X\_i\; sim\; Gammaleft(sum\_\{i=1\}^nk\_i,thetaright).$

If $X$ is a multivariate PDF, then its characteristic function is defined as

- $$

Here, the dot signifies vector dot product ($t$ is in the dual space of $x$).

- $$

If $X$ is a matrix-valued PDF, then the characteristic function is

- $$

varphi_X(T)=Eleft(e^{i, mathrm{Tr}(XT)}right)

Here $mathrm\{Tr\}(cdot)$ is the trace function and matrix multiplication (of $T$ and $X$) is used. Note that the order of the multiplication is immaterial ($XTneq\; TX$ but $tr(XT)=tr(TX)$).

Examples of matrix-valued PDFs include the Wishart distribution.

The characteristic function is closely related to the Fourier transform: the characteristic function of a probability density function $p(x)$ is the complex conjugate of the continuous Fourier transform of $p(x)$ (according to the usual convention; see ).

- $varphi\_X(t)\; =\; langle\; e^\{itX\}\; rangle\; =\; int\_\{-infty\}^\{infty\}\; e^\{itx\}p(x),\; dx\; =\; overline\{left(int\_\{-infty\}^\{infty\}\; e^\{-itx\}p(x),\; dx\; right)\}\; =\; overline\{P(t)\},$

where $P(t)$ denotes the continuous Fourier transform of the probability density function $p(x)$. Likewise, $p(x)$ may be recovered from $varphi\_X(t)$ through the inverse Fourier transform:

- $p(x)\; =\; frac\{1\}\{2pi\}\; int\_\{-infty\}^\{infty\}\; e^\{itx\}\; P(t),\; dt\; =\; frac\{1\}\{2pi\}\; int\_\{-infty\}^\{infty\}\; e^\{itx\}\; overline\{varphi\_X(t)\},\; dt.$

Indeed, even when the random variable does not have a density, the characteristic function may be seen as the Fourier transform of the measure corresponding to the random variable.

- Lukacs E. (1970) Characteristic Functions. Griffin, London. pp. 350
- Bisgaard, T. M., Sasvári, Z. (2000) Characteristic Functions and Moment Sequences, Nova Science

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Last updated on Friday September 26, 2008 at 11:15:19 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 26, 2008 at 11:15:19 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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