Characteristic function (probability theory) explained

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.The characteristic function,

\varphi _{X(t)} = \operatorname{E} \left [ e ^{itX} \right ], a function of,determines the behavior and properties of the probability distribution of the random variable .It is equivalent to a probability density function or cumulative distribution function in the sense that knowing one of the functions it is always possible to find the others, 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.M _{X(t)} , then the domain of the characteristic function can be extended to the complex plane, and\varphi _{X(-it)} = M _{X(t).}

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

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 the characteristic function is defined as the expected value of, where is the imaginary unit, and is the argument of the characteristic function:

\begin{cases} \displaystyle \varphi _{X:R\toC} \ \displaystyle \varphi _{X(t)} = \operatorname{E}\left[e ^{itX} \right] = \int _{R} e ^{itx} dF _{X(x)} = \int _{R} e ^{itx} f _{X(x)dx} = \int ^{1} _{0} e ^{it Q _{X(p)}} dp \end{cases}

Here is the cumulative distribution function of, is the corresponding probability density function, is the corresponding inverse cumulative distribution function also called the quantile function,^[1] and the integrals are of the Riemann–Stieltjes kind. If a random variable has a probability density function then the characteristic function is its Fourier transform with sign reversal in the complex exponential. This convention for the constants appearing in the definition of the characteristic function differs from the usual convention for the Fourier transform. For example, some authors define, which is essentially a change of parameter. Other notation may be encountered in the literature: $\backslash scriptstyle\backslash hat\; p$ as the characteristic function for a probability measure, or $\backslash scriptstyle\backslash hat\; f$ as the characteristic function corresponding to a density .

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 takes its values. For common cases such definitions are listed below:

• If is a -dimensional random vector, then for $$\backslash varphi\_X(t)\; =\; \backslash operatorname\backslash left[\backslash exp(i\; t^T\backslash !X)\backslash right],$$ where $t^T$ is the transpose of the vector  $t$,
• If is a -dimensional random matrix, then for $$\backslash varphi\_X(t)\; =\; \backslash operatorname\backslash left[\backslash exp\; \backslash left(i\; \backslash operatorname\{tr\}(t^T\backslash !X)\; \backslash right)\backslash right],$$ where $\backslash operatorname(\backslash cdot)$ is the trace operator,
• If is a complex random variable, then for $$\backslash varphi\_X(t)\; =\; \backslash operatorname\backslash left[\backslash exp\backslash left(i\; \backslash operatorname\{Re\}\backslash left(\backslash overline\{t\}X\backslash right)\; \backslash right)\backslash right],$$ where $\backslash overline\; t$ is the complex conjugate of $t$ and $\backslash operatorname(z)$ is the real part of the complex number $z$,
• If is a -dimensional complex random vector, then for    $$\backslash varphi\_X(t)\; =\; \backslash operatorname\backslash left[\backslash exp(i\backslash operatorname\{Re\}(t^*\backslash !X))\backslash right],$$ where $t^*$ is the conjugate transpose of the vector  $t$,
• If is a stochastic process, then for all functions such that the integral $\backslash int\_\; t(s)X(s)\backslash ,\backslash mathrms$ converges for almost all realizations of $$\backslash varphi\_X(t)\; =\; \backslash operatorname\backslash left[\backslash exp\; \backslash left\; (i\backslash int\_\backslash mathbf\{R\}\; t(s)X(s)\; \backslash ,\; ds\; \backslash right)\; \backslash right].$$

Examples

Distribution	Characteristic function \phi(t)
	e ^{ita}
	1- p + pe ^{it}
	( 1- p + pe ^{it} ) ^{n}
	\left( (p)/(1- e ^{it} + pe ^{it}) \right) ^{r}
	e ^{λ (e ^{it} -1 )}
	(e ^{itb} - e ^{ita})/(it(b - a))
	(e ^{ita} - e ^{it(b})/(( 1- e ^{it} )(b - a +1 ))
	(e ^{it\mu})/(1+ b ^{2t} ^{2})
Logistic	e ^{i\mu} (\pi s t)/(\sinh(\pi s t))
	e ^{it\mu - (1)/(2) \sigma ^{2t} ^{2}}
	( 1 - 2 it) ^{-k/2}
Noncentral chi-squared {\chi' ^{2} _{k}}	e ^{(i λ t)/(1-2 it)} ( 1 - 2 it) ^{-k/2}
Generalized chi-squared \tilde{\chi}(\boldsymbol{w}, \boldsymbol{k}, \boldsymbol{ λ },s,m)	(\exp\left[it \left( m + \sum _{j} (w _{j} λ _{j})/(1-2 i w _{j} t) \right) - (s ^{2} t ^{2})/(2) \right])/(\prod _{j} \left( 1-2 i w _{j} t \right) ^{k _{j/2}})
	e ^{it\mu}
	( 1 - it\theta) ^{-k}
	( 1 - it λ ^{-1} ) ^{-1}
Geometric (number of failures)	(p)/(1- e ^{it} ( 1- p))
Geometric (number of trials)	(p)/(e ^{-it} - ( 1- p))
	e ^{i{ t ^{T} \boldsymbol{\mu} }-\frac \mathbf^\boldsymbol \mathbf}
Multivariate Cauchy ^[2]	e ^{i t ^{T} \boldsymbol\mu - \sqrt{ t ^{T} \boldsymbol{\Sigma} 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: .
• It is bounded: .
• It is Hermitian: . 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,, both have the same probability distribution if and only if

\varphi _{X _{1}} = \varphi _{X _{2}} .

• If a random variable has moments up to -th order, then the characteristic function is times continuously differentiable on the entire real line. In this case $$\backslash operatorname[X^k]\; =\; i^\; \backslash varphi\_X^(0).$$
• If a characteristic function has a -th derivative at zero, then the random variable has all moments up to if is even, but only up to if is odd. $$\backslash varphi\_X^(0)\; =\; i^k\; \backslash operatorname[X^k]$$
• If are independent random variables, and are some constants, then the characteristic function of the linear combination of the variables is $$\backslash varphi\_(t)\; =\; \backslash varphi\_(a\_1t)\backslash cdots\; \backslash varphi\_(a\_nt).$$ One specific case is the sum of two independent random variables and in which case one has $$\backslash varphi\_(t)\; =\; \backslash varphi\_(t)\backslash cdot\backslash varphi\_(t).$$
• Let

X and Y be two random variables with characteristic functions \varphi _{X} and \varphi _{Y} . X and Y are independent if and only if \varphi _{X,} (s, t) = \varphi _{X} (s) \varphi _{Y} (t) forall (s, t) \in 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 \varphi ^{itb} _{Y(t) = e} \varphi _{X(at)} . For random vectors X and Y = AX + B (where is a constant matrix and a constant vector), we have \varphi _{Y(t)} = e ^{it ^{\top} B} \varphi ^{\top} _{X(A} t) .^[3]

Continuity

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

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

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 when we know the distribution function (or density). If, on the other hand, we know the characteristic function and want to find the corresponding distribution function, then one of the following inversion theorems can be used.

Theorem. If the characteristic function of a random variable is integrable, then is absolutely continuous, and therefore has a probability density function. In the univariate case (i.e. when is scalar-valued) the density function is given by$$f\_X(x)\; =\; F\_X\text{'}(x)\; =\; \backslash frac\backslash int\_\; e^\backslash varphi\_X(t)\backslash ,dt.$$

In the multivariate case it is$$f\_X(x)\; =\; \backslash frac\; \backslash int\_\; e^\backslash varphi\_X(t)\backslash lambda(dt)$$

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

The density function is the Radon–Nikodym derivative of the distribution with respect to the Lebesgue measure :$$f\_X(x)\; =\; \backslash frac(x).$$

Theorem (Lévy). If is characteristic function of distribution function, two points are such that is a continuity set of (in the univariate case this condition is equivalent to continuity of at points and), then

• If is scalar: $$F\_X(b)\; -\; F\_X(a)\; =\; \backslash frac\; \backslash lim\_\; \backslash int\_^\; \backslash frac\; \backslash ,\; \backslash varphi\_X(t)\backslash ,\; dt.$$ This formula can be re-stated in a form more convenient for numerical computation as $$\backslash frac\; =\; \backslash frac\; \backslash int\_^\; \backslash frac\; e^\; \backslash varphi\_X(t)\; \backslash ,\; dt\; .$$ 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 -inf ty gives F(b) , but is numerically impractical.

• If is a vector random variable: $$\backslash mu\_X\backslash big(\backslash \backslash big)\; =\; \backslash frac\; \backslash lim\_\backslash cdots\backslash lim\_\; \backslash int\backslash limits\_\; \backslash cdots\; \backslash int\backslash limits\_\; \backslash prod\_^n\backslash left(\backslash frac\backslash right)\backslash varphi\_X(t)\backslash lambda(dt\_1\; \backslash times\; \backslash cdots\; \backslash times\; dt\_n)$$

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

• If is scalar: $$F\_X(a)\; -\; F\_X(a-0)\; =\; \backslash lim\_\backslash frac\; \backslash int\_^\; e^\backslash varphi\_X(t)\backslash ,dt$$
• If is a vector random variable: $$\backslash mu\_X(\backslash )\; =\; \backslash lim\_\backslash cdots\backslash lim\_\; \backslash left(\backslash prod\_^n\backslash frac\backslash right)\; \backslash int\backslash limits\_\; e^\backslash varphi\_X(t)\backslash lambda(dt)$$

Theorem (Gil-Pelaez). For a univariate random variable, if is a continuity point of then

F _{X(x)} = (1)/(2) - (1)/(\pi) \int ^{inf ty} _{0} (\operatorname{Im)/([e) ^{-itx} \varphi _{X(t)]}{t}dt} where the imaginary part of a complex number z is given by Im (z) = (z - z ^{*)/2i} .

And its density function is:

f _{X(x)} = (1)/(\pi) \int ^{inf ty} _{0} \operatorname{Re}[e ^{-itx} \varphi _{X(t)]dt} The integral may be not Lebesgue-integrable; for example, when is the discrete random variable that is always 0, it becomes the Dirichlet integral.

Inversion formulas for multivariate distributions are available.

Criteria for characteristic functions

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

• A convex linear combination $\backslash sum\_n\; a\_n\backslash varphi\_n(t)$ (with $a\_n\backslash geq0,\backslash \; \backslash sum\_n\; a\_n=1$) 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 is a characteristic function and is a real number, then

\bar{\varphi} ,, and are also characteristic functions.

It is well known that any non-decreasing càdlàg function with limits, corresponds to a cumulative distribution function of some random variable. There is also interest in finding similar simple criteria for when a given function 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.

Bochner’s theorem. An arbitrary function is the characteristic function of some random variable if and only if is positive definite, continuous at the origin, and if .

Khinchine’s criterion. A complex-valued, absolutely continuous function, with, is a characteristic function if and only if it admits the representation

\varphi(t) = \int _{R} g(t + \theta)\overline{g(\theta)} d\theta .

Mathias’ theorem. A real-valued, even, continuous, absolutely integrable function, with, is a characteristic function if and only if

( -1 ) ^{n} \left ( \int _{R} \varphi(pt)e ^{- t ^{2/2}} H _{2n} (t) dt \right ) \geq 0 for, and all . Here denotes the Hermite polynomial of degree .

Pólya’s theorem. If

\varphi is a real-valued, even, continuous function which satisfies the conditions\varphi( 0 ) = 1 ,\varphi is convex for t> 0 ,\varphi( inf ty) = 0 ,then 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,, ..., is a sequence of independent (and not necessarily identically distributed) random variables, and

S _{n} = \sum ^{n} _{i =1} a _{i} X _{i,}

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

\varphi _{S _{n}} (t) = \varphi _{X _{1}} (a _{1t)\varphi} ^{X _{2}} (a _{2t) …} \varphi _{X _{n}} (a _{nt)}

In particular, . To see this, write out the definition of characteristic function:

\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)}

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

Another special case of interest for identically distributed random variables is when and then S_n is the sample mean. In this case, writing for the mean,

\varphi _{\overline{X} }(t)= \varphi_X\!\left(\tfrac \right)^n

Moments

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

$$\backslash operatorname\backslash left[X^n\backslash right]\; =\; i^\backslash left[\backslash frac\{d^n\}\{dt^n\}\backslash varphi\_X(t)\backslash right]\_\; =\; i^\backslash varphi\_X^(0),\backslash !$$

This can be formally written using the derivatives of the Dirac delta function:$$f\_X(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\backslash delta^(x)\backslash operatorname[X^n]$$which allows a formal solution to the moment problem.For example, suppose has a standard Cauchy distribution. Then . This is not differentiable at, showing that the Cauchy distribution has no expectation. Also, the characteristic function of the sample mean of independent observations has characteristic function, 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 follows a Gaussian distribution i.e.

X \sim l{N}(\mu,\sigma ^{2)} . Then \varphi _{X} (t) = e ^{\mu i t - (1)/(2) \sigma ^{2} t ^{2}} and\operatorname{E}\left[ X\right] = i ^{-1} \left[ (d)/(dt) \varphi _{X(t)\right]} _{t=0} = i ^{-1} \left[(i \mu - \sigma ^{2} t) \varphi _{X(t)} \right] _{t=0} = \mu

A similar calculation shows

\operatorname{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 \operatorname{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) and Heathcote (1977) provide some theoretical background for such an estimation procedure. In addition, Yu (2004) 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) and Li et al. (2020) for training generative adversarial networks.

Example

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

( 1 - \theta i t) ^{-k} . Now suppose that we haveX ~\sim \Gamma(k _{1,\theta)} and Y \sim \Gamma(k _{2,\theta)} with and independent from each other, and we wish to know what the distribution of is. The characteristic functions are\varphi _{X(t)=(1} - \theta i t) ^{- k _{1}} , \varphi _{Y(t)=(1} - \theta it) ^{- k _{2}} which by independence and the basic properties of characteristic function leads to\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)}} . This is the characteristic function of the gamma distribution scale parameter and shape parameter, and we therefore concludeX + Y \sim \Gamma(k _{1+k} _{2,\theta)} The result can be expanded to independent gamma distributed random variables with the same scale parameter and we get\forall i \in \{ 1 ,\ldots, n\} : X _{i} \sim \Gamma(k _{i,\theta)} ⇒ \sum ^{n} _{i =1} X _{i} \sim \Gamma\left(\sum ^{nk} _{i,\theta\right).}

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.

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 is the complex conjugate of the continuous Fourier transform of (according to the usual convention; see continuous Fourier transform – other conventions).

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

where denotes the continuous Fourier transform of the probability density function . Likewise, may be recovered from through the inverse Fourier transform:

p(x) = (1)/(2 \pi) \int _{R} e ^{itx} P(t) dt = (1)/(2 \pi) \int _{R} 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.

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.

References

Sources

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• Book: Billingsley, Patrick . Probability and measure . John Wiley & Sons . 1995 . 978-0-471-00710-4 . 3rd.
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• Paulson . A.S. . Holcomb . E.W. . Leitch . R.A. . 1975 . The estimation of the parameters of the stable laws . . 62 . 1 . 163–170 . 10.1093/biomet/62.1.163.
• Book: Pinsky, Mark . Introduction to Fourier analysis and wavelets . Brooks/Cole . 2002 . 978-0-534-37660-4.
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• Wendel . J.G. . 1961 . The non-absolute convergence of Gil-Pelaez' inversion integral . The Annals of Mathematical Statistics . 32 . 1 . 338–339 . 10.1214/aoms/1177705164 . free.
• Yu . J. . 2004 . Empirical characteristic function estimation and its applications . Econometric Reviews . 23 . 2 . 93–1223 . 10.1081/ETC-120039605 . 9076760.
• Shephard . N. G. . 1991a . From characteristic function to distribution function: A simple framework for the theory . Econometric Theory . 7 . 4 . 519–529 . 10.1017/s0266466600004746 . 14668369.
• Shephard . N. G. . 1991b . Numerical integration rules for multivariate inversions . Journal of Statistical Computation and Simulation. 39 . 1–2 . 37–46 . 10.1080/00949659108811337.
• Ansari . Abdul Fatir . Scarlett . Jonathan . Soh . Harold . 2020 . A Characteristic Function Approach to Deep Implicit Generative Modeling . 7478–7487 . Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
• Li . Shengxi . Yu . Zeyang . Xiang . Min . Mandic . Danilo . 2020 . Reciprocal Adversarial Learning via Characteristic Functions . Advances in Neural Information Processing Systems 33 (NeurIPS 2020).

External links

Notes and References

1. 0903.1592 . q-fin.CP . W. T. . Shaw . J. . McCabe . Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space . 2009.
2. using 1 as the number of degree of freedom to recover the Cauchy distribution
3. Web site: Joint characteristic function . 7 April 2018 . www.statlect.com.