Characteristic function (probability theory): Difference between revisions

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{{Short description|Fourier transform of the probability density function}}

[[File:Sinc simple.svg|frame|200px|right|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.]]

[[File:Sinc simple.svg|thumb|280px|right|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 function]]s or [[cumulative distribution function]]s. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

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

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== Introduction ==

The characteristic function providesis an alternativea way forto describingdescribe a [[random variable]] {{mvar|X}}. Similar to the [[cumulative distribution function]],

The '''characteristic function''',

:<math>F_X(x) = \operatorname{E} \left [\mathbf{1}_{\{X\leq x\}} \right]</math>

(where '''1'''_{{''X ≤ x''}} is the [[indicator function]] — it is equal to 1 when {{nowrap|''X ≤ x''}}, and zero otherwise), which completely determines the behavior and properties of the probability distribution of the random variable ''X''. The '''characteristic function''',

: <math> \varphi_X(t) = \operatorname{E} \left [ e^{itX} \right ],</math>

a function of {{mvar|t}},

<!-- What is t ? not defined -->

determines the behavior and properties of the probability distribution of {{mvar|X}}.

also completely determines the behavior and properties of the probability distribution of the random variable ''X''. The two approaches are equivalent in the sense that knowing one of the functions it is always possible to find the other, yet they provide different insights for understanding the features of the random variable. Moreover, in particular cases, there can be differences in whether these functions can be represented as expressions involving simple standard functions.

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 [[probability density function|density function]], then the characteristic function is its [[Duality (mathematics)|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]] <math>M_X(t)</math>, then the ___domain of the characteristic function can be extended to the complex plane, and

: <math> \varphi_X(-it) = M_X(t). </math><ref>{{sfnp|Lukacs (|1970) |p. =196</ref>}}

Note however that the characteristic function of a distribution alwaysis exists,well evendefined whenfor theall [[probabilityreal densitynumber|real functionvalues]] orof {{mvar|t}}, even when the [[moment-generating function]] dois not well defined for all real values of {{mvar|t}}.

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 [[Indecomposable distribution|decomposability]] of random variables.

== Definition ==

For a scalar random variable ''{{mvar|X''}} the '''characteristic function''' is defined as the [[expected value]] of {{math|''e^itX''}}, where ''{{mvar|i''}} is the [[imaginary unit]], and {{nowrapmath|''t'' ∈ '''R'''}} is the argument of the characteristic function:

:<math>\begin{cases} \displaystyle \varphi_X\!:\mathbb{R}\to\mathbb{C} \\ \displaystyle \varphi_X(t) = \operatorname{E}\left[e^{itX}\right] = \int_{\mathbb{R}} e^{itx}\,dF_X(x) = \int_{\mathbb{R}} e^{itx} f_X(x)\,dx = \int_0^1 e^{it Q_X(p)}\,dp \end{cases}</math>

Here {{math|''F_X''}} is the [[cumulative distribution function]] of {{mvar|X}}, {{math|''f_X''}} is the corresponding [[probability density function]], {{math|''Q_X''(''p'')}} is the corresponding inverse cumulative distribution function also called the [[quantile function]],<ref>{{Cite arXiv |eprint=0903.1592 |class=q-fin.CP |first1=W. T. |last1=Shaw |first2=J. |last2=McCabe |title=Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space |year=2009}}</ref> and the integralintegrals isare of the [[Riemann–Stieltjes integral|Riemann–Stieltjes]] kind. If a random variable ''{{mvar|X''}} has a [[probability density function]] ''f_X'', then the characteristic function is its [[Fourier transform]] with sign reversal in the complex exponential,.<ref>{{harvtxtharvp|Statistical and Adaptive Signal Processing|2005|p=79}}</ref><ref>{{harvtxtsfnp|Billingsley|1995|p=345}}</ref> andThis convention for the lastconstants formulaappearing in parenthesesthe isdefinition valid.of <!--the therecharacteristic isfunction nodiffers formulafrom inthe parentheses..usual convention for the Fourier transform.{{sfnp|Pinsky|2002}} -->For example, some authors{{sfnp|Bochner|1955}} define {{math|''Qφ_X''(''pt'') is the inverse cumulative distribution function of{{=}} E[''Xe''^{−2''πitX''}]}}, alsowhich calledis theessentially [[quantilea function]]change of ''X''parameter.<ref>{{Cite arXiv|last1=ShawOther |first1=W.notation T.may |last2=McCabebe |first2=J.encountered |year=2009in |title=Montethe Carloliterature: sampling<math givenstyle="vertical-align:-.3em">\scriptstyle\hat ap</math> Characteristicas Function:the Quantilecharacteristic Mechanicsfunction infor Momentuma Spaceprobability |eprint=0903.1592measure {{mvar|classp}}, or <math style=q"vertical-align:-fin.CP3em">\scriptstyle\hat }}f</refmath> as the characteristic function corresponding to a density {{mvar|f}}.

This convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform.<ref>{{harvtxt|Pinsky|2002}}</ref> For example, some authors<ref>{{harvtxt|Bochner|1955}}</ref> define {{nowrap|''φ_X''(''t'') {{=}} E''e''^{−2''πitX''}}}, which is essentially a change of parameter. Other notation may be encountered in the literature: <math style="vertical-align:-.3em">\scriptstyle\hat p</math> as the characteristic function for a probability measure ''p'', or <math style="vertical-align:-.3em">\scriptstyle\hat f</math> as the characteristic function corresponding to a density ''f''.

== {{anchor|CF Generalizations}} Generalizations ==

The notion of characteristic functions generalizes to multivariate random variables and more complicated [[random element]]s. The argument of the characteristic function will always belong to the [[continuous dual]] of the space where the random variable ''{{mvar|X''}} takes its values. For common cases such definitions are listed below:

* If ''X'' is a ''k''-dimensional [[random vector]], then for {{nowrap|''t'' ∈ '''R'''^''k''}}

::<math> \varphi_X(t) = \operatorname{E}\left[\exp( i t^T\!X)\right], </math>

: where <math display="inline"> t^T</math> is the [[transpose]] of the vector &nbsp;<math display="inline"> t </math>,

* If ''X'' is a ''k''&nbsp;×&nbsp;''p''-dimensional [[random matrix]], then for {{nowrap|''t'' ∈ '''R'''^''k''×''p''}}

::<math> \varphi_X(t) = \operatorname{E}\left[\exp \left( i \operatorname{tr}(t^T\!X) \right )\right], </math>

: where <math display="inline"> \operatorname{tr}(\cdot) </math> is the [[trace (linear algebra)|trace]] operator,

* If ''X'' is a [[complex random variable]], then for {{nowrap|''t'' ∈ '''C'''}} <ref>{{harvtxt|Andersen|Højbjerre|Sørensen|Eriksen|1995|loc=Definition 1.10}}</ref>

::<math>\varphi_X(t) = \operatorname{E}\left[\exp\left( i \operatorname{Re}\left(\overline{t}X\right) \right)\right], </math>

: where <math display="inline">\overline t</math> is the [[complex conjugate]] of&nbsp;<math display="inline">t</math> and <math display="inline"> \operatorname{Re}(z)</math> is the [[real part]] of the complex number <math display="inline"> z </math>,

* If ''X'' is a ''k''-dimensional [[complex random vector]], then for {{nowrap|''t'' ∈ '''C'''^''k''}}&nbsp;&nbsp;<ref>{{harvtxt|Andersen|Højbjerre|Sørensen|Eriksen|1995|loc=Definition 1.20}}</ref>

:: <math> \varphi_X(t) = \operatorname{E}\left[\exp(i\operatorname{Re}(t^*\!X))\right], </math>

: where <math display="inline"> t^* </math> is the conjugate transpose of the vector &nbsp;<math display="inline"> t</math>,

* If ''{{mvar|X''(''s'')}} is a {{mvar|k}}-dimensional [[stochasticrandom processvector]], then for all{{math|''t'' functions∈ ''t'R'(''s^{'') such that the integralk''}}} <math display="inlineblock"> \int_varphi_X(t) = \operatorname{E}\mathbbleft[\exp( R}i t(s)^T\!X(s)\right],\mathrm{d}s </math> convergeswhere for<math almostdisplay="inline"> allt^T</math> realizationsis the [[transpose]] of ''X''the vector &nbsp;<ref>{{harvtxt|Sobczyk|2001|pagemath display=20}}"inline"> t </refmath>,

::* If {{mvar|X}} is a {{math|''k'' × ''p''}}-dimensional [[random matrix]], then for {{math|''t'' ∈ '''R'''^''k''×''p''}} <math display="block"> \varphi_X(t) = \operatorname{E}\left[\exp \left ( i \int_\mathbfoperatorname{Rtr} t(s)t^T\!X(s) \, ds \right ) \right]., </math> where <math display="inline"> \operatorname{tr}(\cdot) </math> is the [[trace (linear algebra)|trace]] operator,

* If {{mvar|X}} is a [[complex random variable]], then for {{math|''t'' ∈ '''C'''}}{{sfnp|Andersen|Højbjerre|Sørensen|Eriksen|1995|loc=Definition 1.10}} <math display="block">\varphi_X(t) = \operatorname{E}\left[\exp\left( i \operatorname{Re}\left(\overline{t}X\right) \right)\right], </math> where <math display="inline">\overline t</math> is the [[complex conjugate]] of&nbsp;<math display="inline">t</math> and <math display="inline"> \operatorname{Re}(z)</math> is the [[real part]] of the [[complex number]] <math display="inline"> z </math>,

* If {{mvar|X}} is a {{mvar|k}}-dimensional [[complex random vector]], then for {{math|''t'' ∈ '''C'''^''k''}}&nbsp;&nbsp;{{sfnp|Andersen|Højbjerre|Sørensen|Eriksen|1995|loc=Definition 1.20}} <math display="block"> \varphi_X(t) = \operatorname{E}\left[\exp(i\operatorname{Re}(t^*\!X))\right], </math> where <math display="inline"> t^* </math> is the [[conjugate transpose]] of the vector &nbsp;<math display="inline"> t</math>,

* If {{math|''X''(''s'')}} is a [[stochastic process]], then for all functions {{math|''t''(''s'')}} such that the integral <math display="inline"> \int_{\mathbb R} t(s)X(s)\,\mathrm{d}s </math> converges for almost all realizations of {{mvar|X}}{{sfnp|Sobczyk|2001|p=20}} <math display="block">\varphi_X(t) = \operatorname{E}\left[\exp \left ( i\int_\mathbf{R} t(s)X(s) \, ds \right ) \right]. </math>

== Examples ==

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|-

! Distribution

! Characteristic function ''φ''<math>\varphi(''t'')</math>

|-

| [[Degenerate distribution|Degenerate]] {{math|''δ''_''a''}}

| <math>e^{ita}</math>

|-

| [[Bernoulli distribution|Bernoulli]] {{math|Bern(''p'')}}

| <math>1-p+pe^{it}</math>

|-

| [[Binomial distribution|Binomial]] {{math|B(''n, p'')}}

| <math>(1-p+pe^{it})^n</math>

|-

| [[Negative binomial distribution|Negative binomial]] {{math|NB(''r, p'')}}

| <math>\left(\frac{1-p}{1 - p e^{i\,tit} + pe^{it}}\right)^{\!r} </math>

|-

| [[Poisson distribution|Poisson]] {{math|Pois(''λ'')}}

| <math>e^{\lambda(e^{it}-1)}</math>

|-

| [[Uniform distribution (continuous)|Uniform (continuous)]] {{math|U(''a, b'')}}

| <math>\frac{e^{itb} - e^{ita}}{it(b-a)}</math>

|-

|[[Discrete uniform distribution|Uniform (discrete)]] {{math|DU(''a, b'')}}

| <math>\frac{e^{aitita} - e^{it(b + 1)it}}{(b - a + 1)(1 - e^{it})(b - a + 1)}</math>

|-

| [[Laplace distribution|Laplace]] {{math|L(''μ'', ''b'')}}

| <math>\frac{e^{it\mu}}{1 + b^2t^2}</math>

|-

| [[NormalLogistic distribution|NormalLogistic]] {{math|Logistic(''Nμ''(,''μ,s'')}}<br σ²'')

| <math>e^{i\mu t}\frac{\pi s t}{\sinh(\pi s t)}</math>

|-

| [[Normal distribution|Normal]] {{math|''N''(''μ'', ''σ''²)}}

| <math>e^{it\mu - \frac{1}{2}\sigma^2t^2}</math>

|-

| [[Chi-squared distribution|Chi-squared]] {{math|1=''χ''²<sub style="position:relative;left:-5pt;top:2pt">''k''</sub>}}

| <math>(1 - 2it)^{-k/2}</math>

|-

| [[CauchyNoncentral chi-squared distribution|CauchyNoncentral chi-squared]] C(''μ, θ'<math>{\chi')_k}^2</math>

| <math>e^{\frac{i\lambda t}{1-2it}}(1 - 2it)^{-k/2}</math>

|-

| [[Generalized chi-squared distribution|Generalized chi-squared]] <math>\tilde{\chi}(\boldsymbol{w}, \boldsymbol{k}, \boldsymbol{\lambda},s,m)</math>

| <math>\frac{\exp\left[it \left( m + \sum_j \frac{w_j \lambda_j}{1-2i w_j t} \right)-\frac{s^2 t^2}{2}\right]}{\prod_j \left(1-2i w_j t \right)^{k_j/2}}</math>

|-

| [[Cauchy distribution|Cauchy]] {{math|C(''μ'', ''θ'')}}

| <math>e^{it\mu -\theta|t|}</math>

|-

| [[Gamma distribution|Gamma]] {{math|Γ(''k'', ''θ'')}}

| <math>(1 - it\theta)^{-k}</math>

|-

| [[Exponential distribution|Exponential]] {{math|Exp(''λ'')}}

| <math>(1 - it\lambda^{-1})^{-1}</math>

|-

| [[Geometric distribution|Geometric]] {{math|Gf(''p'')}}<br />(number of failures)

| <math>\frac{p}{1-e^{it}(1-p)}</math>

|-

| [[Geometric distribution|Geometric]] {{math|Gt(''p'')}}<br />(number of trials)

| <math>\frac{p}{e^{-it}-(1-p)}</math>

|-

| [[Multivariate normal distribution|Multivariate normal]] {{math|''N''('''''μ''''', '''''Σ''''')}}

| <math>e^{i{ \mathbf{t}^{\mathrm{T}}\left(i{ \boldsymbol{\mu}}-\frac {1}{2} \mathbf{t}^{\mathrm{T}}\boldsymbol{\Sigma} \mathbf{t} \right)} </math>

|-

| [[Multivariate Cauchy distribution|Multivariate Cauchy]] ''{{math|MultiCauchy''('''''μ''''', '''''Σ''''')}}<ref>{{harvp|Kotz et al. |Nadarajah|2004|p. =37}} using 1 as the number of degree of freedom to recover the Cauchy distribution</ref>

| <math>e^{i\mathbf{t}^{\mathrm{T}}\boldsymbol\mu - \sqrt{\mathbf{t}^{\mathrm{T}}\boldsymbol{\Sigma} \mathbf{t}}}</math>

|-

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== 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 (mathematics)|measure]] is finite.

* A characteristic function is [[Uniform continuity|uniformly continuous]] on the entire space.

* It is non-vanishing in a region around zero: {{math|1=''φ''(0) = 1}}.

* It is bounded: {{math|{{abs|''φ''(''t'')|}} ≤ 1}}.

* It is [[Hermitian function|Hermitian]]: {{nowrapmath|''φ''(−''t'') {{=}} {{overline|''φ''(''t'')}}}}. In particular, the characteristic function of a symmetric (around the origin) random variable is real-valued and [[even and odd functions|even]].

* There is a [[bijection]] between [[probability distribution]]s and characteristic functions. That is, for any two random variables {{math|''X''₁}}, {{math|''X''₂}}, both have the same probability distribution if and only if <math> \varphi_{X_1}=\varphi_{X_2}</math>. {{Citation needed|reason=proof ?|date=October 2023}}

* If a random variable ''{{mvar|X''}} has [[Moment (mathematics)|moments]] up to ''{{mvar|k''}}-th order, then the characteristic function {{math|''φ''_''X''}} is ''{{mvar|k''}} times continuously differentiable on the entire real line. In this case <math display="block">\operatorname{E}[X^k] = i^{-k} \varphi_X^{(k)}(0).</math>

* If a characteristic function {{math|''φ''_''X''}} has a {{mvar|k}}-th derivative at zero, then the random variable {{mvar|X}} has all moments up to {{mvar|k}} if {{mvar|k}} is even, but only up to {{math|''k'' – 1}} if {{mvar|k}} is odd.{{sfnp|Lukacs|1970|loc=Corollary 1 to Theorem 2.3.1}} <math display="block"> \varphi_X^{(k)}(0) = i^k \operatorname{E}[X^k] </math>

:: <math>\operatorname{E}[X^k] = i^{-k} \varphi_X^{(k)}(0).</math>

* If {{math|''X''₁, ..., ''X_n''}} are independent random variables, and {{math|''a''₁, ..., ''a_n''}} are some constants, then the characteristic function of the linear combination of the {{math|''X''_''i''}} variables is <math display="block">\varphi_{a_1X_1+\cdots+a_nX_n}(t) = \varphi_{X_1}(a_1t)\cdots \varphi_{X_n}(a_nt).</math> One specific case is the sum of two independent random variables {{math|''X''₁}} and {{math|''X''₂}} in which case one has <math display="block">\varphi_{X_1+X_2}(t) = \varphi_{X_1}(t)\cdot\varphi_{X_2}(t).</math>

* If a characteristic function φ_''X'' has a ''k''-th derivative at zero, then the random variable ''X'' has all moments up to ''k'' if ''k'' is even, but only up to {{nowrap|''k'' – 1}} if ''k'' is odd.<ref>Lukacs (1970), Corollary 1 to Theorem 2.3.1</ref>

* Let <math>X</math> and <math>Y</math> be two random variables with characteristic functions <math>\varphi_{X}</math> and <math>\varphi_{Y}</math>. <math>X</math> and <math>Y</math> are independent if and only if <math>\varphi_{X, Y}(s, t)= \varphi_{X}(s) \varphi_{Y}(t) \quad \text { for all } \quad(s, t) \in \mathbb{R}^{2}</math>.

:: <math> \varphi_X^{(k)}(0) = i^k \operatorname{E}[X^k] </math>

* If ''X''₁, ..., ''X_n'' are independent random variables, and ''a''₁, ..., ''a_n'' are some constants, then the characteristic function of the linear combination of the ''X''_''i'' 's is

:: <math>\varphi_{a_1X_1+\cdots+a_nX_n}(t) = \varphi_{X_1}(a_1t)\cdots \varphi_{X_n}(a_nt).</math>

::One specific case is the sum of two independent random variables ''X''₁ and ''X''₂ in which case one has

:: <math>\varphi_{X_1+X_2}(t)=\varphi_{X_1}(t)\cdot\varphi_{X_2}(t).</math>

* The tail behavior of the characteristic function determines the [[smoothness (probability theory)|smoothness]] of the corresponding density function.

* Let the random variable <math>Y = aX + b</math> be the linear transformation of a random variable <math>X</math>. The characteristic function of <math>Y</math> is <math>\varphi_Y(t)=e^{itb}\varphi_X(at)</math>. For random vectors <math>X</math> and <math>Y = AX + B</math> (where ''{{mvar|A''}} is a constant matrix and ''{{mvar|B''}} a constant vector), we have <math>\varphi_Y(t) = e^{it^\top B}\varphi_X(A^\top t)</math>.<ref>{{citeCite web |title=Joint characteristic function |url=https://www.statlect.com/fundamentals-of-probability/joint-characteristic-function |titleaccess-date=Joint7 characteristicApril 2018 function|website=www.statlect.com|access-date=7 April 2018}}</ref>

=== Continuity ===

The bijection stated above between probability distributions and characteristic functions is ''sequentially continuous''. That is, whenever a sequence of distribution functions {{math|''F_j''(''x'') }} converges (weakly) to some distribution {{math|''F''(''x'')}}, the corresponding sequence of characteristic functions {{math|''φ''_''j''(''t'')}} will also converge, and the limit {{math|''φ''(''t'')}} will correspond to the characteristic function of law ''{{mvar|F''}}. More formally, this is stated as

: '''[[Lévy’s continuity theorem]]:''' A sequence {{math|''X_j''}} of ''{{mvar|n''}}-variate random variables [[Convergence in distribution|converges in distribution]] to random variable ''{{mvar|X''}} if and only if the sequence {{math|''φ''_{''X_j''}}} converges pointwise to a function {{mvar|φ}} which is continuous at the origin. Where {{mvar|φ}} is the characteristic function of ''{{mvar|X''}}.<ref>{{harvtxtsfnp|Cuppens|1975|loc=Theorem 2.6.9}}</ref>

This theorem can be used to prove the [[Law of large numbers#Proof using convergence of characteristic functions|law of large numbers]] and the [[Central limit theorem#Proof|central limit theorem]].

=== Inversion formulaeformula ===

There is a [[Bijection|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 ''{{mvar|φ''}} when we know the distribution function ''{{mvar|F''}} (or density ''{{mvar|f''}}). If, on the other hand, we know the characteristic function ''{{mvar|φ''}} and want to find the corresponding distribution function, then one of the following '''inversion theorems''' can be used.

'''Theorem'''. If the characteristic function {{math|''φ_X''}} of a random variable ''{{mvar|X''}} is [[Integrable function|integrable]], then {{math|''F_X''}} is absolutely continuous, and therefore ''{{mvar|X''}} has a [[probability density function]]. In the univariate case (i.e. when ''{{mvar|X''}} is scalar-valued) the density function is given by

<math display="block"> f_X(x) = F_X'(x) = \frac{1}{2\pi}\int_{\mathbf{R}} e^{-itx}\varphi_X(t)\,dt.</math>

: <math> f_X(x) = F_X'(x) = \frac{1}{2\pi}\int_{\mathbf{R}} e^{-itx}\varphi_X(t)\,dt.</math>

In the multivariate case it is

<math display="block"> f_X(x) = \frac{1}{(2\pi)^n} \int_{\mathbf{R}^n} e^{-i(t\cdot x)}\varphi_X(t)\lambda(dt)</math>

where <math display="inline"> t\cdot x</math> is the [[dot product]].

: <math> f_X(x) = \frac{1}{(2\pi)^n} \int_{\mathbf{R}^n} e^{-i(t\cdot x)}\varphi_X(t)\lambda(dt)</math>

where <math display="inline"> t\cdot x</math> is the dot-product.

The pdf is the [[Radon–Nikodym derivative]] of the distribution ''μ_X'' with respect to the [[Lebesgue measure]] ''λ'':

: <math> f_X(x) = \frac{d\mu_X}{d\lambda}(x). </math>

The density function is the [[Radon–Nikodym derivative]] of the distribution {{math|''μ_X''}} with respect to the [[Lebesgue measure]] {{mvar|λ}}:

'''Theorem (Lévy)'''.{{NoteTag|named after the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]]}} If ''φ''_''X'' is characteristic function of distribution function ''F_X'', two points ''a''&nbsp;<&nbsp;''b'' are such that {{nowrap|{''x'' {{!}} ''a'' < ''x'' < ''b''}}} is a [[continuity set]] of ''μ''_''X'' (in the univariate case this condition is equivalent to continuity of ''F_X'' at points ''a'' and ''b''), then

<math display="block"> f_X(x) = \frac{d\mu_X}{d\lambda}(x). </math>

* If ''X'' is scalar:

::<math>F_X(b) - F_X(a) = \frac{1} {2\pi} \lim_{T \to \infty} \int_{-T}^{+T} \frac{e^{-ita} - e^{-itb}} {it}\, \varphi_X(t)\, dt.</math>

:This formula can be re-stated in a form more convenient for numerical computation as <ref name="auto">Shepard, N.G. (1991a)</ref>

:::<math> \frac{F(x+h) - F(x-h)}{2h} = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{\sin ht}{ht} e^{-itx} \varphi_X(t) \, dt .</math>

:For a random variable bounded from below one can obtain <math>F(b)</math> by taking <math>a</math> such that <math>F(a)=0.</math> Otherwise, if a random variable is not bounded from below, the limit for <math>a\to-\infty</math> gives <math>F(b)</math>, but is numerically impractical.<ref name="auto" />

'''Theorem (Lévy)'''.{{NoteTag|named after the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]]}} If {{math|''φ''_''X''}} is characteristic function of distribution function {{math|''F_X''}}, two points {{math|''a'' < ''b''}} are such that {{math|{{mset|''x'' {{!}} ''a'' < ''x'' < ''b''}}}} is a [[continuity set]] of {{math|''μ''_''X''}} (in the univariate case this condition is equivalent to continuity of {{math|''F_X''}} at points {{mvar|a}} and {{mvar|b}}), then

* If ''X'' is a vector random variable:

* If {{mvar|X}} is scalar: <math display="block">F_X(b) - F_X(a) = \frac{1} {2\pi} \lim_{T \to \infty} \int_{-T}^{+T} \frac{e^{-ita} - e^{-itb}} {it}\, \varphi_X(t)\, dt.</math> This formula can be re-stated in a form more convenient for numerical computation as{{sfnp|Shephard|1991a}} <math display="block"> \frac{F(x+h) - F(x-h)}{2h} = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{\sin ht}{ht} e^{-itx} \varphi_X(t) \, dt .</math> For a random variable bounded from below one can obtain <math>F(b)</math> by taking <math>a</math> such that <math>F(a)=0.</math> Otherwise, if a random variable is not bounded from below, the limit for <math>a\to-\infty</math> gives <math>F(b)</math>, but is numerically impractical.{{sfnp|Shephard|1991a}}

::<math>\mu_X\big(\{a<x<b\}\big) = \frac{1}{(2\pi)^n} \lim_{T_1\to\infty}\cdots\lim_{T_n\to\infty} \int\limits_{-T_1\leq t_1\leq T_1} \cdots \int\limits_{-T_n \leq t_n \leq T_n} \prod_{k=1}^n\left(\frac{e^{-it_ka_k}-e^{-it_kb_k}}{it_k}\right)\varphi_X(t)\lambda(dt_1 \times \cdots \times dt_n)</math>

* If {{mvar|X}} is a vector random variable: <math display="block">\mu_X\big(\{a<x<b\}\big) = \frac{1}{(2\pi)^n} \lim_{T_1\to\infty}\cdots\lim_{T_n\to\infty} \int\limits_{-T_1\leq t_1\leq T_1} \cdots \int\limits_{-T_n \leq t_n \leq T_n} \prod_{k=1}^n\left(\frac{e^{-i t_k a_k}-e^{-i t_k b_k}}{it_k}\right)\varphi_X(t)\lambda(dt_1 \times \cdots \times dt_n)</math>

'''Theorem'''. If ''{{mvar|a''}} is (possibly) an atom of ''{{mvar|X''}} (in the univariate case this means a point of discontinuity of {{math|''F_X'' }}) then

* If {{mvar|X}} is scalar: <math display="block">F_X(a) - F_X(a-0) = \lim_{T\to\infty}\frac{1}{2T} \int_{-T}^{+T} e^{-ita}\varphi_X(t)\,dt</math>

* If ''X'' is scalar:

* If {{mvar|X}} is a vector random variable::{{sfnp|Cuppens|1975|loc=Theorem 2.3.2}} <math display="block">F_X\mu_X(\{a) - F_X(a-0\}) = \lim_{TT_1\to\infty}\cdots\lim_{T_n\to\infty} \left(\prod_{k=1}^n\frac{1}{2T2T_k}\int_right) \int\limits_{[-T}^{+TT_1, T_1] \times \dots \times [-T_n, T_n]} e^{-itai(t\cdot a)}\varphi_X(t)\,lambda(dt)</math>

* If ''X'' is a vector random variable:<ref>{{harvtxt|Cuppens|1975|loc=Theorem 2.3.2}}</ref>

:: <math>\mu_X(\{a\}) = \lim_{T_1\to\infty}\cdots\lim_{T_n\to\infty} \left(\prod_{k=1}^n\frac{1}{2T_k}\right) \int\limits_{[-T_1, T_1] \times \dots \times [-T_n, T_n]} e^{-i(t\cdot a)}\varphi_X(t)\lambda(dt)</math>

'''Theorem (Gil-Pelaez)'''.<ref>{{sfnp|Wendel, J.G. (|1961)</ref>}} For a univariate random variable ''{{mvar|X''}}, if ''{{mvar|x''}} is a [[continuity point]] of {{math|''F_X''}} then

: <math>F_X(x) = \frac{1}{2} - \frac{1}{\pi}\int_0^\infty \frac{\operatorname{Im}[e^{-itx}\varphi_X(t)]}{t}\,dt.</math>

where the imaginary part of a complex number <math>z</math> is given by <math>\mathrm{Im}(z) = (z - z^*)/2i</math>.

And its density function is:

The integral may be not [[Lebesgue-integrable]]; for example, when ''X'' is the [[discrete random variable]] that is always 0, it becomes the [[Dirichlet integral]].

: <math>f_X(x) = \frac{1}{\pi}\int_0^\infty \operatorname{Re}[e^{-itx}\varphi_X(t)]\,dt</math>

The integral may be not [[Lebesgue-integrable]]; for example, when {{mvar|X}} is the [[discrete random variable]] that is always 0, it becomes the [[Dirichlet integral]].

Inversion formulas for multivariate distributions are available.<ref>{{sfnp|Shephard (|1991a,b)</ref>}}{{sfnp|Shephard|1991b}}

=== Criteria for characteristic functions ===

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

*A [[convex combination|convex linear combination]] <math display="inline"> \sum_n a_n\varphi_n(t)</math> (with <math display="inline"> a_n\geq0,\ \sum_n a_n=1</math>) 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 ''{{mvar|φ''}} is a characteristic function and {{mvar|α}} is a real number, then <math>\bar{\varphi}</math>, {{math|Re(''φ''), {{abs|''φ''|}}²}}, and {{math|''φ''(''αt'')}} are also characteristic functions.

It is well known that any non-decreasing [[càdlàg]] function ''{{mvar|F''}} with limits {{math|1=''F''(−∞) = 0}}, {{math|1=''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 ''{{mvar|φ''}} could be the characteristic function of some random variable. The central result here is [[Bochner's theorem|Bochner’s theorem]], although its usefulness is limited because the main condition of the theorem, [[positive -definite function|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[[George Pólya|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.<ref>{{sfnp|Lukacs (|1970), |p. =84</ref>}}

'''[[Bochner's theorem|Bochner’s theorem]]'''. An arbitrary function {{math|''φ'' : '''R'''^''n'' → '''C'''}} is the characteristic function of some random variable if and only if ''{{mvar|φ''}} is [[positive -definite function|positive definite]], continuous at the origin, and if {{math|1=''φ''(0) = 1}}.

'''Khinchine’s criterion'''. A complex-valued, absolutely continuous function ''{{mvar|φ''}}, with {{math|1=''φ''(0) = 1}}, is a characteristic function if and only if it admits the representation

: <math>\varphi(t) = \int_{\mathbf{R}} g(t+\theta)\overline{g(\theta)} \, d\theta .</math>

'''Mathias’ theorem'''. A real-valued, even, continuous, absolutely integrable function ''{{mvar|φ''}}, with {{math|1=''φ''(0) = 1}}, is a characteristic function if and only if

:<math>(-1)^n \left ( \int_{\mathbf{R}} \varphi(pt)e^{-t^2/2} H_{2n}(t) \, dt \right ) \geq 0</math>

for {{math|1=''n'' = 0,1,2,...}}, and all {{math|''p'' > 0}}. Here {{math|''H''_2''n''}} denotes the [[Hermite polynomials|Hermite polynomial]] of degree {{math|2''n''}}.

[[File:2 cfs coincide over a finite interval.svg|thumb|250px|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.]]

Line 202 ⟶ 184:

* <math> \varphi </math> is [[convex function|convex]] for <math> t>0 </math>,

* <math> \varphi(\infty) = 0 </math>,

then {{math|''φ''(''t'')}} is the characteristic function of an absolutely continuous distribution symmetric about 0.

== Uses ==

Line 208 ⟶ 190:

=== Basic manipulations of distributions ===

Characteristic functions are particularly useful for dealing with linear functions of [[statistical independence|independent]] random variables. For example, if {{nowrapmath|''X''₁}}, {{math|''X''₂}}, ..., {{math|''X_n''}} is a sequence of independent (and not necessarily identically distributed) random variables, and

:<math>S_n = \sum_{i=1}^n a_i X_i,\,\!</math>

where the {{math|''a''_''i''}} are constants, then the characteristic function for {{math|''S''_''n''}} is given by

:<math>\varphi_{S_n}(t)=\varphi_{X_1}(a_1t)\varphi_{X_2}(a_2t)\cdots \varphi_{X_n}(a_nt) \,\!</math>

In particular, {{nowrapmath|''φ_X+Y''(''t'') {{=}} ''φ_X''(''t'')''φ_Y''(''t'')}}. To see this, write out the definition of characteristic function:

: <math>\varphi_{X+Y}(t)= \operatorname{E}\left [e^{it(X+Y)}\right]= \operatorname{E}\left [e^{itX}e^{itY}\right] = \operatorname{E}\left [e^{itX}\right] \operatorname{E}\left [e^{itY}\right] =\varphi_X(t) \varphi_Y(t)</math>

The independence of ''{{mvar|X''}} and ''{{mvar|Y''}} is required to establish the equality of the third and fourth expressions.

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

: <math>\varphi_{\overline{X}}(t)= \varphi_X\!\left(\tfrac{t}{n} \right)^n</math>

=== Moments ===

Characteristic functions can also be used to find [[moment (mathematics)|moments]] of a random variable. Provided that the ''{{mvar|n''}}-^th moment exists, the characteristic function can be differentiated ''{{mvar|n''}} times and:

<math display=block>

:<math> \left[\frac{d^n}{dt^n} \varphi_X(t)\right]_{t=0} = i^{n} \operatorname{E}\left[ X^n\right] \Rightarrow

\operatorname{E}\left[ X^n\right] = i^{-n}\left[\frac{d^n}{dt^n}\varphi_X(t)\right]_{t=0} = i^{-n}\varphi_X^{(n)}(0) ,\!</math>

This can be formally written using the derivatives of the [[Dirac delta function]]:<math display="block">f_X(x) = \sum_{n=0}^\infty \frac{(-1)^n}{n!}\delta^{(n)}(x)\operatorname{E}[X^n]

For example, suppose ''X'' has a standard [[Cauchy distribution]]. Then {{nowrap|''φ_X''(''t'') {{=}} ''e''^{−{{!}}''t''{{!}}}}}. This is not [[Differentiable function|differentiable]] at ''t'' = 0, showing that the Cauchy distribution has no [[expected value|expectation]]. Also, the characteristic function of the sample mean {{overline|''X''}} of ''n'' [[Statistical independence|independent]] observations has characteristic function {{nowrap|''φ''_{{{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.

</math>which allows a formal solution to the [[moment problem]].

For example, suppose {{mvar|X}} has a standard [[Cauchy distribution]]. Then {{math|''φ_X''(''t'') {{=}} ''e''^{−{{!}}''t''{{!}}}}}. This is not [[Differentiable function|differentiable]] at {{math|1=''t'' = 0}}, showing that the Cauchy distribution has no [[expected value|expectation]]. Also, the characteristic function of the sample mean {{math|{{overline|''X''}}}} of {{mvar|n}} [[Statistical independence|independent]] observations has characteristic function {{math|''φ''_{{{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.

As a further example, suppose ''{{mvar|X''}} follows a [[Gaussian distribution]] i.e. <math>X \sim \mathcal{N}(\mu,\sigma^2)</math>. Then <math>\varphi_{X}(t) = e^{\mu i t - \frac{1}{2} \sigma^2 t^2} </math> and

:<math>\operatorname{E}\left[ X\right] = i^{-1} \left[\frac{d^n}{dt^n}\varphi_X(t)\right]_{t=0} = i^{-1} \left[(i \mu - \sigma^2 t) \varphi_X(t) \right]_{t=0} = \mu </math>

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

Line 243 ⟶ 227:

=== 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)<ref>{{harvtxtsfnp|Paulson| Holcomb | Leitch |1975}}</ref> and Heathcote (1977)<ref>{{harvtxtsfnp|Heathcote|1977}}</ref> provide some theoretical background for such an estimation procedure. In addition, Yu (2004)<ref>{{harvtxtsfnp|Yu|2004}}</ref> 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)<ref>{{harvtxtsfnp|Ansari| Scarlett |Soh|2020}}</ref> and Li et al. (2020)<ref>{{harvtxtsfnp|Li| Yu | Xiang | Mandic|2020}}</ref> for training [[generative adversarial networks]].

=== Example ===

The [[gamma distribution]] with scale parameter θ and a shape parameter ''{{mvar|k''}} has the characteristic function

: <math>(1 - \theta\, i\, t)^{-k}.</math>

Now suppose that we have

: <math> X ~\sim \Gamma(k_1,\theta) \mbox{ and } Y \sim \Gamma(k_2,\theta) \,</math>

with ''{{mvar|X''}} and ''{{mvar|Y''}} independent from each other, and we wish to know what the distribution of {{math|''X'' + ''Y''}} is. The characteristic functions are

: <math>\varphi_X(t)=(1 - \theta\, i\, t)^{-k_1},\,\qquad \varphi_Y(t)=(1 - \theta\,i\,t it)^{-k_2}</math>

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

: <math>\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\, t\right)^{-(k_1+k_2)}.</math>

This is the characteristic function of the gamma distribution scale parameter ''{{mvar|θ''}} and shape parameter {{math|''k''₁ + ''k''₂}}, and we therefore conclude

: <math>X+Y \sim \Gamma(k_1+k_2,\theta) \,</math>

The result can be expanded to ''{{mvar|n''}} independent gamma distributed random variables with the same scale parameter and we get

: <math>\forall i \in \{1,\ldots, n\} : X_i \sim \Gamma(k_i,\theta) \qquad \Rightarrow \qquad \sum_{i=1}^n X_i \sim \Gamma\left(\sum_{i=1}^nk_i,\theta\right).</math>

== Entire characteristic functions ==

{{Expand section|date=December 2009}}

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|analyticalanalytic continuation]], in cases where this is possible.<ref>{{harvtxtsfnp|Lukacs|1970|loc=Chapter 7}}</ref>

== 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 {{math|''p''(''x'')}} is the [[complex conjugate]] of the [[continuous Fourier transform]] of {{math|''p''(''x'')}} (according to the usual convention; see [[Continuous Fourier transform#Other conventions|continuous Fourier transform – other conventions]]).

: <math>\varphi_X(t) = \langle e^{itX} \rangle = \int_{\mathbf{R}} e^{itx}p(x)\, dx = \overline{\left( \int_{\mathbf{R}} e^{-itx}p(x)\, dx \right)} = \overline{P(t)},</math>

where {{math|''P''(''t'')}} denotes the [[continuous Fourier transform]] of the probability density function {{math|''p''(''x'')}}. Likewise, {{math|''p''(''x'')}} may be recovered from {{math|''φ_X''(''t'')}} through the inverse Fourier transform:

:<math>p(x) = \frac{1}{2\pi} \int_{\mathbf{R}} e^{itx} P(t)\, dt = \frac{1}{2\pi} \int_{\mathbf{R}} e^{itx} \overline{\varphi_X(t)}\, dt.</math>

Line 281 ⟶ 265:

Below was lifted from [[generating function]] ... there should be an analog for the characteristic function

*Suppose that ''{{mvar|N''}} is also an independent, discrete random variable taking values on the non-negative integers, with probability-generating function {{math|''G''_''N''}}. If the {{math|''X''₁}}, {{math|''X''₂}}, ..., {{math|''X''_''N''}} are independent ''and'' identically distributed with common probability-generating function {{math|''G''_''X''}}, then

::<math>G_{S_N}(z) = G_N(G_X(z)).</math>

Line 301 ⟶ 285:

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* {{cite journal | last = Shephard | first = N. G. | year = 1991b | title = Numerical integration rules for multivariate inversions | url = https://ora.ox.ac.uk/objects/uuid:da00666a-4790-4666-a54c-b81fc6fc49cb| journal = J. Statist. Comput. Simul | volume = 39 | issue = 1–2| pages = 37–46 | doi=10.1080/00949659108811337 }}

* {{cite conference

| url = https://openaccess.thecvf.com/content_CVPR_2020/html/Ansari_A_Characteristic_Function_Approach_to_Deep_Implicit_Generative_Modeling_CVPR_2020_paper.html

| title = A Characteristic Function Approach to Deep Implicit Generative Modeling

| last1 = Ansari

| first1 = Abdul Fatir

| last2 = Scarlett

| first2 = Jonathan

| last3 = Soh

| first3 = Harold

| year = 2020

| book-title = Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2020

| pages = 7478-7487

}}

* {{cite conference

| url = https://proceedings.neurips.cc/paper/2020/hash/021f6dd88a11ca489936ae770e4634ad-Abstract.html

| title = Reciprocal Adversarial Learning via Characteristic Functions

| last1 = Li

| first1 = Shengxi

| last2 = Yu

| first2 = Zeyang

| last3 = Xiang

| first3 = Min

| last4 = Mandic

| first4 = Danilo

| year = 2020

| book-title = Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

}}

{{refend}}