
In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics, the characteristic function of any
real-valued
In mathematics, value may refer to several, strongly related notions.
In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an ...
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
completely defines its
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
. If a random variable admits a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
, then the characteristic function is the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
s or
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
s. 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
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
. 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 provides an alternative way for describing a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
. Similar to the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
,
:
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
— it is equal to 1 when , and zero otherwise), which completely determines the behavior and properties of the probability distribution of the random variable ''X''. The characteristic function,
:
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.
If a random variable admits a
density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
, then the characteristic function is its
Fourier dual, in the sense that each of them is a
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the other. If a random variable has a
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
, then the domain of the characteristic function can be extended to the complex plane, and
:
Note however that the characteristic function of a distribution always exists, even when the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
or
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
do not.
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...
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 ''X'' the characteristic function is defined as the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of ''e
itX'', where ''i'' is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
, and is the argument of the characteristic function:
:
Here ''F
X'' is the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of ''X'', and the integral is of the
Riemann–Stieltjes kind. If a random variable ''X'' has a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
''f
X'', then the characteristic function is its
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
with sign reversal in the complex exponential. ''Q
X''(''p'') is the inverse cumulative distribution function of ''X'' also called the
quantile function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value e ...
of ''X''.
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:
as the characteristic function for a probability measure ''p'', or
as the characteristic function corresponding to a density ''f''.
Generalizations
The notion of characteristic functions generalizes to multivariate random variables and more complicated
random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansio ...
s. The argument of the characteristic function will always belong to the
continuous dual of the space where the random variable ''X'' takes its values. For common cases such definitions are listed below:
* If ''X'' is a ''k''-dimensional
random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
, then for
where
is the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of the vector
,
* If ''X'' is a ''k'' × ''p''-dimensional
random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...
, then for
where
is the
trace operator,
* If ''X'' is a
complex random variable In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can a ...
, then for
where
is the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of
and
is the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
of the complex number
,
* If ''X'' is a ''k''-dimensional
complex random vector In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If Z_1,\ldots,Z_n are com ...
, then for
where
is the conjugate transpose of the vector
,
* If ''X''(''s'') is a
stochastic process, then for all functions ''t''(''s'') such that the integral
converges for almost all realizations of ''X''
Examples
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
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. I ...
on the entire space
* It is non-vanishing in a region around zero: φ(0) = 1.
* It is bounded: , φ(''t''), ≤ 1.
* It is
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
: . In particular, the characteristic function of a symmetric (around the origin) random variable is real-valued and
even.
* There is a
bijection between
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s and characteristic functions. That is, for any two random variables ''X''
1, ''X''
2, both have the same probability distribution if and only if
.
* If a random variable ''X'' has
moments up to ''k''-th order, then the characteristic function φ
''X'' is ''k'' times continuously differentiable on the entire real line. In this case
* 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 if ''k'' is odd.
* 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
One specific case is the sum of two independent random variables ''X''
1 and ''X''
2 in which case one has
* Let
and
be two random variables with characteristic functions
and
.
and
are independent if and only if
.
* The tail behavior of the characteristic function determines the
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
of the corresponding density function.
* Let the random variable
be the linear transformation of a random variable
. The characteristic function of
is
. For random vectors
and
(where ''A'' is a constant matrix and ''B'' a constant vector), we have
.
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 ''F''. More formally, this is stated as
:
Lévy’s continuity theorem: A sequence ''X
j'' of ''n''-variate random variables
converges in distribution to random variable ''X'' if and only if the sequence φ
''Xj'' converges pointwise to a function φ which is continuous at the origin. Where φ is the characteristic function of ''X''.
This theorem can be used to prove the
law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials sho ...
and the
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...
.
Inversion formula
There is a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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 ''φ'' when we know the distribution function ''F'' (or density ''f''). If, on the other hand, we know the characteristic function ''φ'' 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 ''X'' is
integrable, then ''F
X'' is absolutely continuous, and therefore ''X'' has a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
. In the univariate case (i.e. when ''X'' is scalar-valued) the density function is given by
In the multivariate case it is
where
is the dot-product.
The pdf is the
Radon–Nikodym derivative of the distribution ''μ
X'' with respect to the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
''λ'':
Theorem (Lévy). If ''φ''
''X'' is characteristic function of distribution function ''F
X'', two points ''a'' < ''b'' are such that is a
continuity set In measure theory, a branch of mathematics, a continuity set of a measure ''μ'' is any Borel set ''B'' such that
: \mu(\partial B) = 0\,,
where \partial B is the (topological) boundary of ''B''. For signed measures, one asks that
: , \mu, ( ...
of ''μ''
''X'' (in the univariate case this condition is equivalent to continuity of ''F
X'' at points ''a'' and ''b''), then
* If ''X'' is scalar:
This formula can be re-stated in a form more convenient for numerical computation as
For a random variable bounded from below one can obtain
by taking
such that
Otherwise, if a random variable is not bounded from below, the limit for
gives
, but is numerically impractical.
* If ''X'' is a vector random variable:
Theorem. If ''a'' is (possibly) an atom of ''X'' (in the univariate case this means a point of discontinuity of ''F
X'' ) then
* If ''X'' is scalar:
* If ''X'' is a vector random variable:
Theorem (Gil-Pelaez). For a univariate random variable ''X'', if ''x'' is a
continuity point
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
of ''F
X'' then
:
where the imaginary part of a complex number
is given by
.
The integral may be not
Lebesgue-integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
; for example, when ''X'' is the
discrete random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
that is always 0, it becomes the
Dirichlet integral
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
: \int_0^\inft ...
.
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 (with
) 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 ''φ'' is a characteristic function and α is a real number, then
, Re(''φ''), , ''φ'',
2, and ''φ''(''αt'') are also characteristic functions.
It is well known that any non-decreasing
càdlàg function ''F'' with limits ''F''(−∞) = 0, ''F''(+∞) = 1 corresponds to a
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of some random variable. There is also interest in finding similar simple criteria for when a given function ''φ'' 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 ''φ'' : R
''n'' → C is the characteristic function of some random variable if and only if ''φ'' is
positive definite, continuous at the origin, and if ''φ''(0) = 1.
Khinchine’s criterion. A complex-valued, absolutely continuous function ''φ'', with ''φ''(0) = 1, is a characteristic function if and only if it admits the representation
:
Mathias’ theorem. A real-valued, even, continuous, absolutely integrable function ''φ'', with ''φ''(0) = 1, is a characteristic function if and only if
:
for ''n'' = 0,1,2,..., and all ''p'' > 0. Here ''H''
2''n'' denotes the
Hermite polynomial of degree 2''n''.

Pólya’s theorem. If
is a real-valued, even, continuous function which satisfies the conditions
*
,
*
is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
for
,
*
,
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
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...
. 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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
random variables. For example, if is a sequence of independent (and not necessarily identically distributed) random variables, and
:
where the ''a''
''i'' are constants, then the characteristic function for ''S''
''n'' is given by
:
In particular, . To see this, write out the definition of characteristic function:
:
The independence of ''X'' and ''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 and then ''S
n'' is the sample mean. In this case, writing for the mean,
:
Moments
Characteristic functions can also be used to find
moments of a random variable. Provided that the ''n''
th moment exists, the characteristic function can be differentiated ''n'' times and
:
For example, suppose ''X'' has a standard
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fu ...
. Then . This is not
differentiable at ''t'' = 0, showing that the Cauchy distribution has no
expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* ''Expectation' ...
. Also, the characteristic function of the sample mean of ''n''
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
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 ''X'' follows a
Gaussian distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
i.e.
. Then
and
:
A similar calculation shows
and is easier to carry out than applying the definition of expectation and using integration by parts to evaluate
.
The logarithm of a characteristic function is a
cumulant generating function, which is useful for finding
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will hav ...
s; some instead define the cumulant generating function as the logarithm of the
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
, 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
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...
since closed form expressions for the density are not available which makes implementation of
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed sta ...
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
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. E ...
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
A generative adversarial network (GAN) is a class of machine learning frameworks designed by Ian Goodfellow and his colleagues in June 2014. Two neural networks contest with each other in the form of a zero-sum game, where one agent's gain is a ...
.
Example
The
gamma distribution
In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
with scale parameter θ and a shape parameter ''k'' has the characteristic function
:
Now suppose that we have
:
with ''X'' and ''Y'' independent from each other, and we wish to know what the distribution of ''X'' + ''Y'' is. The characteristic functions are
:
which by independence and the basic properties of characteristic function leads to
:
This is the characteristic function of the gamma distribution scale parameter ''θ'' and shape parameter ''k''
1 + ''k''
2, and we therefore conclude
:
The result can be expanded to ''n'' independent gamma distributed random variables with the same scale parameter and we get
:
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
analytical continuation, in cases where this is possible.
Related concepts
Related concepts include the
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
and the
probability-generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are oft ...
. 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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
: the characteristic function of a probability density function ''p''(''x'') is the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the
continuous Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of ''p''(''x'') (according to the usual convention; see
continuous Fourier transform – other conventions).
:
where ''P''(''t'') denotes the
continuous Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the probability density function ''p''(''x''). Likewise, ''p''(''x'') may be recovered from ''φ
X''(''t'') through the inverse Fourier transform:
:
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
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g i ...
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 In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving ...
.
See also
*
Subindependence, a weaker condition than independence, that is defined in terms of characteristic functions.
*
Cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will hav ...
, a term of the ''cumulant generating functions'', which are logs of the characteristic functions.
Notes
References
Citations
Sources
*
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External links
*
{{Theory of probability distributions
Functions related to probability distributions