Characteristic function (probability theory)

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The characteristic function of a uniform U(–1,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued.

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases.

The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function.

Introduction

The characteristic function is a way to describe a random variable Template:Mvar. The characteristic function,

${\phi }_X(t)=\mathrm{E}\left[e^{itX}\right],$

a function of Template:Mvar, determines the behavior and properties of the probability distribution of Template:Mvar. It is equivalent to a probability density function or cumulative distribution function, since knowing one of these functions allows computation of the others, but they provide different insights into the features of the random variable. In particular cases, one or another of these equivalent functions may be easier to represent in terms of simple standard functions.

If a random variable admits a density function, then the characteristic function is its Fourier dual, in the sense that each of them is a Fourier transform of the other. If a random variable has a moment-generating function $M_X(t)$, then the domain of the characteristic function can be extended to the complex plane, and

${\phi }_X(-it)=M_X(t).$Template:Sfnp

Note however that the characteristic function of a distribution is well defined for all real values of Template:Mvar, even when the moment-generating function is not well defined for all real values of Template:Mvar.

The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.

Definition

For a scalar random variable Template:Mvar the characteristic function is defined as the expected value of e^itX, where Template:Mvar is the imaginary unit, and t ∈ R is the argument of the characteristic function:

$\{\begin{array}{l}{\phi }_X:\mathbb{ℝ}\to \mathbb{ℂ}\\ {\phi }_X(t)=\mathrm{E}\left[e^{itX}\right]=\underset{\mathbb{ℝ}}{\int }{e}^{itx}d{F}_X(x)=\underset{\mathbb{ℝ}}{\int }{e}^{itx}{f}_X(x)dx=\underset{0}{\overset{1}{\int }}{e}^{it{Q}_X(p)}dp\end{array}$

Here F_X is the cumulative distribution function of Template:Mvar, f_X is the corresponding probability density function, Q_X(p) is the corresponding inverse cumulative distribution function also called the quantile function, and the integrals are of the Riemann–Stieltjes kind. If a random variable Template:Mvar has a probability density function then the characteristic function is its Fourier transform with sign reversal in the complex exponential.Template:Sfnp This convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform.Template:Sfnp For example, some authorsTemplate:Sfnp define φ_X(t) = E[e^−2πitX], which is essentially a change of parameter. Other notation may be encountered in the literature: $\hat{p}$ as the characteristic function for a probability measure Template:Mvar, or $\hat{f}$ as the characteristic function corresponding to a density Template:Mvar.

Generalizations

The notion of characteristic functions generalizes to multivariate random variables and more complicated random elements. The argument of the characteristic function will always belong to the continuous dual of the space where the random variable Template:Mvar takes its values. For common cases such definitions are listed below:

• If Template:Mvar is a Template:Mvar-dimensional random vector, then for t ∈ R^k $${\phi }_X(t)=\mathrm{E}[\exp (i{t}^{T}X)],$$ where $t^T$ is the transpose of the vector  $t$,
• If Template:Mvar is a k × p-dimensional random matrix, then for t ∈ R^k×p $${\phi }_X(t)=\mathrm{E}[\exp (i\mathrm{tr}(t^{T}X))],$$ where $\mathrm{tr}(\cdot )$ is the trace operator,
• If Template:Mvar is a complex random variable, then for t ∈ CTemplate:Sfnp $${\phi }_X(t)=\mathrm{E}[\exp (i\mathrm{Re}\left(\bar{t}X\right))],$$ where $\bar{t}$ is the complex conjugate of $t$ and $\mathrm{Re}(z)$ is the real part of the complex number $z$,
• If Template:Mvar is a Template:Mvar-dimensional complex random vector, then for t ∈ C^k  Template:Sfnp $${\phi }_X(t)=\mathrm{E}[\exp (i\mathrm{Re}(t^{*}X))],$$ where $t^{*}$ is the conjugate transpose of the vector  $t$,
• If X(s) is a stochastic process, then for all functions t(s) such that the integral $\underset{\mathbb{ℝ}}{\int }t(s)X(s)\mathrm{d}s$ converges for almost all realizations of Template:MvarTemplate:Sfnp $${\phi }_X(t)=\mathrm{E}[\exp (i\underset{\mathbf{𝐑}}{\int }t(s)X(s)ds)].$$

Examples

Distribution	Characteristic function $\phi (t)$
Degenerate δa	$e^{ita}$
Bernoulli Bern(p)	$1-p+p{e}^{it}$
Binomial B(n, p)	$(1-p+p{e}^{it}{)}^n$
Negative binomial NB(r, p)	${\left(\frac{p}{1-e^{it}+p{e}^{it}}\right)}^r$
Poisson Pois(λ)	$e^{\lambda (e^{it}-1)}$
Uniform (continuous) U(a, b)	$\frac{e^{itb}-e^{ita}}{it(b-a)}$
Uniform (discrete) DU(a, b)	$\frac{e^{ita}-e^{it(b+1)}}{(1-e^{it})(b-a+1)}$
Laplace L(μ, b)	$\frac{e^{it\mu }}{1+b^2{t}^2}$
Logistic Logistic(μ,s)	$e^{i\mu t}\frac{\pi st}{\sinh (\pi st)}$
Normal N(μ, σ2)	$e^{it\mu -\frac{1}{2}{\sigma }^2{t}^2}$
Chi-squared χ2k	$(1-2it{)}^{-k/2}$
Noncentral chi-squared ${\chi {\text{'}}_k}^2$	$e^{\frac{i\lambda t}{1-2it}}(1-2it{)}^{-k/2}$
Generalized chi-squared $\tilde{\chi }(\mathit{w},\mathit{k},\mathit{\lambda },s,m)$	$\frac{\exp [it(m+\sum _j\frac{w_j{\lambda }_j}{1-2i{w}_{j}t})-\frac{s^2{t}^2}{2}]}{\prod _j{(1-2i{w}_{j}t)}^{k_j/2}}$
Cauchy C(μ, θ)	$e^{it\mu -\theta |t|}$
Gamma Γ(k, θ)	$(1-it\theta {)}^{-k}$
Exponential Exp(λ)	$(1-it{\lambda }^{-1}{)}^{-1}$
Geometric Gf(p) (number of failures)	$\frac{p}{1-e^{it}(1-p)}$
Geometric Gt(p) (number of trials)	$\frac{p}{e^{-it}-(1-p)}$
Multivariate normal N(μ, Σ)	$e^{i{\mathbf{𝐭}}^{\mathrm{T}}\mathit{\mu }-\frac{1}{2}{\mathbf{𝐭}}^{\mathrm{T}}\mathrm{\Sigma }\mathbf{𝐭}}$
Multivariate Cauchy MultiCauchy(μ, Σ)	$e^{i{\mathbf{𝐭}}^{\mathrm{T}}\mathit{\mu }-\sqrt{{\mathbf{𝐭}}^{\mathrm{T}}\mathrm{\Sigma }\mathbf{𝐭}}}$

Oberhettinger (1973) provides extensive tables of characteristic functions.

Properties

• The characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite.
• A characteristic function is uniformly continuous on the entire space.
• It is non-vanishing in a region around zero: φ(0) = 1.
• It is bounded: Template:Abs ≤ 1.
• It is Hermitian: φ(−t) = φ(t). In particular, the characteristic function of a symmetric (around the origin) random variable is real-valued and even.
• There is a bijection between probability distributions and characteristic functions. That is, for any two random variables X₁, X₂, both have the same probability distribution if and only if ${\phi }_{X_1}={\phi }_{X_2}$.
• If a random variable Template:Mvar has moments up to Template:Mvar-th order, then the characteristic function φ_X is Template:Mvar times continuously differentiable on the entire real line. In this case $$\mathrm{E}[X^k]=i^{-k}{\phi }_X^{(k)}(0).$$
• If a characteristic function φ_X has a Template:Mvar-th derivative at zero, then the random variable Template:Mvar has all moments up to Template:Mvar if Template:Mvar is even, but only up to k – 1 if Template:Mvar is odd.Template:Sfnp $${\phi }_X^{(k)}(0)=i^k\mathrm{E}[X^k]$$
• If X₁, ..., X_n are independent random variables, and a₁, ..., a_n are some constants, then the characteristic function of the linear combination of the X_i variables is $${\phi }_{a_1{X}_1+\cdots +a_n{X}_n}(t)={\phi }_{X_1}(a_{1}t)\cdots {\phi }_{X_n}(a_{n}t).$$ One specific case is the sum of two independent random variables X₁ and X₂ in which case one has $${\phi }_{X_1+X_2}(t)={\phi }_{X_1}(t)\cdot {\phi }_{X_2}(t).$$
• Let $X$ and $Y$ be two random variables with characteristic functions ${\phi }_X$ and ${\phi }_Y$. $X$ and $Y$ are independent if and only if ${\phi }_{X,Y}(s,t)={\phi }_X(s){\phi }_Y(t)\text{ for all }(s,t)\in {\mathbb{ℝ}}^2$.
• The tail behavior of the characteristic function determines the smoothness of the corresponding density function.
• Let the random variable $Y=aX+b$ be the linear transformation of a random variable $X$. The characteristic function of $Y$ is ${\phi }_Y(t)=e^{itb}{\phi }_X(at)$. For random vectors $X$ and $Y=AX+B$ (where Template:Mvar is a constant matrix and Template:Mvar a constant vector), we have ${\phi }_Y(t)=e^{i{t}^{\mathrm{\top }}B}{\phi }_X(A^{\mathrm{\top }}t)$.

Continuity

The bijection stated above between probability distributions and characteristic functions is sequentially continuous. That is, whenever a sequence of distribution functions F_j(x) converges (weakly) to some distribution F(x), the corresponding sequence of characteristic functions φ_j(t) will also converge, and the limit φ(t) will correspond to the characteristic function of law Template:Mvar. More formally, this is stated as

Lévy’s continuity theorem: A sequence X_j of Template:Mvar-variate random variables converges in distribution to random variable Template:Mvar if and only if the sequence φ_Xj converges pointwise to a function Template:Mvar which is continuous at the origin. Where Template:Mvar is the characteristic function of Template:Mvar.Template:Sfnp

This theorem can be used to prove the law of large numbers and the central limit theorem.

Inversion formula

There is a one-to-one correspondence between cumulative distribution functions and characteristic functions, so it is possible to find one of these functions if we know the other. The formula in the definition of characteristic function allows us to compute Template:Mvar when we know the distribution function Template:Mvar (or density Template:Mvar). If, on the other hand, we know the characteristic function Template:Mvar and want to find the corresponding distribution function, then one of the following inversion theorems can be used.

Theorem. If the characteristic function φ_X of a random variable Template:Mvar is integrable, then F_X is absolutely continuous, and therefore Template:Mvar has a probability density function. In the univariate case (i.e. when Template:Mvar is scalar-valued) the density function is given by $$f_X(x)={F_X}^{\prime }(x)=\frac{1}{2\pi }\underset{\mathbf{𝐑}}{\int }{e}^{-itx}{\phi }_X(t)dt.$$

In the multivariate case it is $$f_X(x)=\frac{1}{(2\pi {)}^n}\underset{{\mathbf{𝐑}}^n}{\int }{e}^{-i(t\cdot x)}{\phi }_X(t)\lambda (dt)$$

where $t\cdot x$ is the dot product.

The density function is the Radon–Nikodym derivative of the distribution μ_X with respect to the Lebesgue measure Template:Mvar: $$f_X(x)=\frac{d{\mu }_X}{d\lambda }(x).$$

Theorem (Lévy).Template:NoteTag If φ_X is characteristic function of distribution function F_X, two points a < b are such that Template:Mset is a continuity set of μ_X (in the univariate case this condition is equivalent to continuity of F_X at points Template:Mvar and Template:Mvar), then

• If Template:Mvar is scalar: $$F_X(b)-F_X(a)=\frac{1}{2\pi }\underset{T\to \mathrm{\infty }}{\lim }\underset{-T}{\overset{+T}{\int }}\frac{e^{-ita}-e^{-itb}}{it}{\phi }_X(t)dt.$$ This formula can be re-stated in a form more convenient for numerical computation asTemplate:Sfnp $$\frac{F(x+h)-F(x-h)}{2h}=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sin ht}{ht}{e}^{-itx}{\phi }_X(t)dt.$$ For a random variable bounded from below one can obtain $F(b)$ by taking $a$ such that $F(a)=0.$ Otherwise, if a random variable is not bounded from below, the limit for $a\to -\mathrm{\infty }$ gives $F(b)$, but is numerically impractical.Template:Sfnp
• If Template:Mvar is a vector random variable: $${\mu }_X(\{a<x<b\})=\frac{1}{(2\pi {)}^n}\underset{T_1\to \mathrm{\infty }}{\lim }\cdots \underset{T_n\to \mathrm{\infty }}{\lim }\underset{-T_1\le t_1\le T_1}{\int }\cdots \underset{-T_n\le t_n\le T_n}{\int }\prod _{k=1}^n\left(\frac{e^{-i{t}_k{a}_k}-e^{-i{t}_k{b}_k}}{i{t}_k}\right){\phi }_X(t)\lambda (d{t}_1\times \cdots \times d{t}_n)$$

Theorem. If Template:Mvar is (possibly) an atom of Template:Mvar (in the univariate case this means a point of discontinuity of F_X) then

• If Template:Mvar is scalar: $$F_X(a)-F_X(a-0)=\underset{T\to \mathrm{\infty }}{\lim }\frac{1}{2T}\underset{-T}{\overset{+T}{\int }}{e}^{-ita}{\phi }_X(t)dt$$
• If Template:Mvar is a vector random variable:Template:Sfnp $${\mu }_X(\{a\})=\underset{T_1\to \mathrm{\infty }}{\lim }\cdots \underset{T_n\to \mathrm{\infty }}{\lim }\left(\prod _{k=1}^n\frac{1}{2T_k}\right)\underset{[-T_1,T_1]\times \dots \times [-T_n,T_n]}{\int }{e}^{-i(t\cdot a)}{\phi }_X(t)\lambda (dt)$$

Theorem (Gil-Pelaez).Template:Sfnp For a univariate random variable Template:Mvar, if Template:Mvar is a continuity point of F_X then

$F_X(x)=\frac{1}{2}-\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\mathrm{Im}[e^{-itx}{\phi }_X(t)]}{t}dt$

where the imaginary part of a complex number $z$ is given by $\mathrm{I}\mathrm{m}(z)=(z-z^{*})/2i$.

And its density function is:

$f_X(x)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{Re}[e^{-itx}{\phi }_X(t)]dt$

The integral may be not Lebesgue-integrable; for example, when Template:Mvar is the discrete random variable that is always 0, it becomes the Dirichlet integral.

Inversion formulas for multivariate distributions are available.Template:SfnpTemplate:Sfnp

Criteria for characteristic functions

The set of all characteristic functions is closed under certain operations:

• A convex linear combination $\sum _n{a}_n{\phi }_n(t)$ (with $a_n\ge 0,\sum _n{a}_n=1$) of a finite or a countable number of characteristic functions is also a characteristic function.
• The product of a finite number of characteristic functions is also a characteristic function. The same holds for an infinite product provided that it converges to a function continuous at the origin.
• If Template:Mvar is a characteristic function and Template:Mvar is a real number, then $\overline{\phi }$, Re(φ), Template:Abs², and φ(αt) are also characteristic functions.

It is well known that any non-decreasing càdlàg function Template:Mvar with limits F(−∞) = 0, F(+∞) = 1 corresponds to a cumulative distribution function of some random variable. There is also interest in finding similar simple criteria for when a given function Template:Mvar could be the characteristic function of some random variable. The central result here is Bochner’s theorem, although its usefulness is limited because the main condition of the theorem, non-negative definiteness, is very hard to verify. Other theorems also exist, such as Khinchine’s, Mathias’s, or Cramér’s, although their application is just as difficult. Pólya’s theorem, on the other hand, provides a very simple convexity condition which is sufficient but not necessary. Characteristic functions which satisfy this condition are called Pólya-type.Template:Sfnp

Bochner’s theorem. An arbitrary function φ : Rⁿ → C is the characteristic function of some random variable if and only if Template:Mvar is positive definite, continuous at the origin, and if φ(0) = 1.

Khinchine’s criterion. A complex-valued, absolutely continuous function Template:Mvar, with φ(0) = 1, is a characteristic function if and only if it admits the representation

$\phi (t)=\underset{\mathbf{𝐑}}{\int }g(t+\theta )\overline{g(\theta )}d\theta .$

Mathias’ theorem. A real-valued, even, continuous, absolutely integrable function Template:Mvar, with φ(0) = 1, is a characteristic function if and only if

$(-1{)}^n(\underset{\mathbf{𝐑}}{\int }\phi (pt)e^{-t^2/2}{H}_{2n}(t)dt)\ge 0$

for n = 0,1,2,..., and all p > 0. Here H_2n denotes the Hermite polynomial of degree 2n.

File:2 cfs coincide over a finite interval.svg

Pólya’s theorem can be used to construct an example of two random variables whose characteristic functions coincide over a finite interval but are different elsewhere.

Pólya’s theorem. If $\phi$ is a real-valued, even, continuous function which satisfies the conditions

• $\phi (0)=1$,
• $\phi$ is convex for $t>0$,
• $\phi (\mathrm{\infty })=0$,

then φ(t) is the characteristic function of an absolutely continuous distribution symmetric about 0.

Uses

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

Basic manipulations of distributions

Characteristic functions are particularly useful for dealing with linear 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

$S_n=\sum _{i=1}^n{a}_i{X}_i,$

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

${\phi }_{S_n}(t)={\phi }_{X_1}(a_{1}t){\phi }_{X_2}(a_{2}t)\cdots {\phi }_{X_n}(a_{n}t)$

In particular, φ_X+Y(t) = φ_X(t)φ_Y(t). To see this, write out the definition of characteristic function:

${\phi }_{X+Y}(t)=\mathrm{E}\left[e^{it(X+Y)}\right]=\mathrm{E}\left[e^{itX}{e}^{itY}\right]=\mathrm{E}\left[e^{itX}\right]\mathrm{E}\left[e^{itY}\right]={\phi }_X(t){\phi }_Y(t)$

The independence of Template:Mvar and Template:Mvar is required to establish the equality of the third and fourth expressions.

Another special case of interest for identically distributed random variables is when a_i = 1 / n and then S_n is the sample mean. In this case, writing X for the mean,

${\phi }_{\bar{X}}(t)={\phi }_X{\left(\frac{t}{n}\right)}^n$

Moments

Characteristic functions can also be used to find moments of a random variable. Provided that the Template:Mvar-^th moment exists, the characteristic function can be differentiated Template:Mvar times:

$$\mathrm{E}\left[X^n\right]=i^{-n}{[\frac{d^n}{d{t}^n}{\phi }_X(t)]}_{t=0}=i^{-n}{\phi }_X^{(n)}(0),$$

This can be formally written using the derivatives of the Dirac delta function:$$f_X(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{n!}{\delta }^{(n)}(x)\mathrm{E}[X^n]$$which allows a formal solution to the moment problem. For example, suppose Template:Mvar has a standard Cauchy distribution. Then φ_X(t) = e^−|t|. This is not differentiable at t = 0, showing that the Cauchy distribution has no expectation. Also, the characteristic function of the sample mean X of Template:Mvar independent observations has characteristic function φ_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.

As a further example, suppose Template:Mvar follows a Gaussian distribution i.e. $X\sim {𝒩}(\mu ,{\sigma }^2)$. Then ${\phi }_X(t)=e^{\mu it-\frac{1}{2}{\sigma }^2{t}^2}$ and

$\mathrm{E}\left[X\right]=i^{-1}{[\frac{d}{dt}{\phi }_X(t)]}_{t=0}=i^{-1}{[(i\mu -{\sigma }^{2}t){\phi }_X(t)]}_{t=0}=\mu$

A similar calculation shows $\mathrm{E}\left[X^2\right]={\mu }^2+{\sigma }^2$ and is easier to carry out than applying the definition of expectation and using integration by parts to evaluate $\mathrm{E}\left[X^2\right]$.

The logarithm of a characteristic function is a cumulant generating function, which is useful for finding cumulants; some instead define the cumulant generating function as the logarithm of the moment-generating function, and call the logarithm of the characteristic function the second cumulant generating function.

Data analysis

Characteristic functions can be used as part of procedures for fitting probability distributions to samples of data. Cases where this provides a practicable option compared to other possibilities include fitting the stable distribution since closed form expressions for the density are not available which makes implementation of maximum likelihood estimation difficult. Estimation procedures are available which match the theoretical characteristic function to the empirical characteristic function, calculated from the data. Paulson et al. (1975)Template:Sfnp and Heathcote (1977)Template:Sfnp provide some theoretical background for such an estimation procedure. In addition, Yu (2004)Template:Sfnp describes applications of empirical characteristic functions to fit time series models where likelihood procedures are impractical. Empirical characteristic functions have also been used by Ansari et al. (2020)Template:Sfnp and Li et al. (2020)Template:Sfnp for training generative adversarial networks.

Example

The gamma distribution with scale parameter θ and a shape parameter Template:Mvar has the characteristic function

$(1-\theta it{)}^{-k}.$

Now suppose that we have

$X\sim \mathrm{\Gamma }(k_1,\theta )\text{ and }Y\sim \mathrm{\Gamma }(k_2,\theta )$

with Template:Mvar and Template:Mvar independent from each other, and we wish to know what the distribution of X + Y is. The characteristic functions are

${\phi }_X(t)=(1-\theta it{)}^{-k_1},{\phi }_Y(t)=(1-\theta it{)}^{-k_2}$

which by independence and the basic properties of characteristic function leads to

${\phi }_{X+Y}(t)={\phi }_X(t){\phi }_Y(t)=(1-\theta it{)}^{-k_1}(1-\theta it{)}^{-k_2}={(1-\theta it)}^{-(k_1+k_2)}.$

This is the characteristic function of the gamma distribution scale parameter Template:Mvar and shape parameter k₁ + k₂, and we therefore conclude

$X+Y\sim \mathrm{\Gamma }(k_1+k_2,\theta )$

The result can be expanded to Template:Mvar independent gamma distributed random variables with the same scale parameter and we get

$\mathrm{\forall }i\in \{1,\dots ,n\}:X_i\sim \mathrm{\Gamma }(k_i,\theta )\Rightarrow \sum _{i=1}^n{X}_i\sim \mathrm{\Gamma }(\sum _{i=1}^n{k}_i,\theta ).$

Entire characteristic functions

As defined above, the argument of the characteristic function is treated as a real number: however, certain aspects of the theory of characteristic functions are advanced by extending the definition into the complex plane by analytic continuation, in cases where this is possible.Template:Sfnp

Related concepts

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

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 continuous Fourier transform – other conventions).

${\phi }_X(t)=⟨e^{itX}⟩=\underset{\mathbf{𝐑}}{\int }{e}^{itx}p(x)dx=\overline{(\underset{\mathbf{𝐑}}{\int }{e}^{-itx}p(x)dx)}=\overline{P(t)},$

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

$p(x)=\frac{1}{2\pi }\underset{\mathbf{𝐑}}{\int }{e}^{itx}P(t)dt=\frac{1}{2\pi }\underset{\mathbf{𝐑}}{\int }{e}^{itx}\overline{{\phi }_X(t)}dt.$

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.

Another related concept is the representation of probability distributions as elements of a reproducing kernel Hilbert space via the kernel embedding of distributions. This framework may be viewed as a generalization of the characteristic function under specific choices of the kernel function.

See also

• Subindependence, a weaker condition than independence, that is defined in terms of characteristic functions.
• Cumulant, a term of the cumulant generating functions, which are logs of the characteristic functions.

External links

• Template:Springer