Characteristic function (probability theory)
In probability theory, the characteristic function of a random variable encodes its probability distribution through the Fourier transform. Every probability distribution possesses a characteristic function, including distributions whose moments or moment-generating functions do not exist. Characteristic functions determine probability laws uniquely and convert sums of independent random variables into products of ordinary complex-valued functions.
For a real-valued random variable (X), its characteristic function is
[ \varphi_X(t) = \operatorname{E}!\left[e^{itX}\right] = \int_{\mathbb R} e^{itx},dF_X(x), \qquad t\in\mathbb R, ]
where (i^2=-1), (\operatorname{E}) denotes expected value, and (F_X) is the cumulative distribution function of (X). Equivalently, if (\mu_X) is the probability measure induced by (X), then
[ \varphi_X(t)=\int_{\mathbb R} e^{itx},\mu_X(dx). ]
The term “characteristic function” also has unrelated meanings in set theory and convex analysis. Within probability theory, it conventionally refers to the Fourier transform of a probability measure.
Definition and immediate properties
The defining expectation always exists because (|e^{itX}|=1). Decomposition into real and imaginary parts gives
[ \varphi_X(t) = \operatorname{E}[\cos(tX)]
- i,\operatorname{E}[\sin(tX)]. ]
Consequently,
[ \varphi_X(0)=1, \qquad |\varphi_X(t)|\leq 1, \qquad \varphi_X(-t)=\overline{\varphi_X(t)}. ]
Every characteristic function is uniformly continuous on (\mathbb R). This follows from dominated convergence together with the identity
[ \varphi_X(t+h)-\varphi_X(t) = \operatorname{E}!\left[e^{itX}(e^{ihX}-1)\right]. ]
A shift and rescaling of the random variable correspond to a phase factor and a change of argument. If (Y=aX+b), then
[ \varphi_Y(t)=e^{itb}\varphi_X(at). ]
When (X) and (Y) are independent random variables, their sum satisfies
[ \varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t). ]
This product relation is central to the treatment of convolutions, sums, and limiting distributions. In measure-theoretic notation, the characteristic function of the convolution (\mu*\nu) equals the pointwise product of the characteristic functions of (\mu) and (\nu).
Positive definiteness
Characteristic functions belong to the class of continuous positive-definite functions. For arbitrary real numbers (t_1,\ldots,t_n) and complex numbers (z_1,\ldots,z_n),
[ \sum_{j=1}^{n}\sum_{k=1}^{n} z_j\overline{z_k}, \varphi_X(t_j-t_k)
\operatorname{E}!\left[ \left| \sum_{j=1}^{n} z_j e^{it_jX} \right|^2 \right] \geq 0. ]
Bochner’s theorem gives the converse characterization: a function (\varphi:\mathbb R\to\mathbb C) is the characteristic function of a probability measure precisely when it is positive definite, continuous at zero, and satisfies (\varphi(0)=1). Continuity at zero, together with positive definiteness, entails continuity on the entire real line.
This characterization distinguishes characteristic functions from arbitrary bounded Fourier transforms. Normalization represents total probability, while positive definiteness records the nonnegative nature of the underlying measure.
Uniqueness and inversion
A probability distribution is uniquely determined by its characteristic function. Thus, if
[ \varphi_X(t)=\varphi_Y(t) \quad\text{for every }t\in\mathbb R, ]
then (X) and (Y) have the same distribution. This remains true even when neither distribution has finite moments.
At continuity points (a<b) of the distribution function (F_X), a Fourier inversion formula takes the form
[ F_X(b)-F_X(a)
\lim_{T\to\infty} \frac{1}{2\pi} \int_{-T}^{T} \frac{e^{-ita}-e^{-itb}}{it}, \varphi_X(t),dt. ]
If the law has a probability density (f_X) and the relevant integrability conditions hold, then the density is recovered through
[ f_X(x)
\frac{1}{2\pi} \int_{\mathbb R} e^{-itx}\varphi_X(t),dt. ]
The exact form of an inversion theorem depends on whether the distribution is continuous, discrete, or mixed. The uniqueness conclusion itself requires no density and no integrability of (\varphi_X).
For an integer-valued random variable (N), the characteristic function is (2\pi)-periodic, and its probability mass function satisfies
[ \Pr(N=k)
\frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-itk}\varphi_N(t),dt. ]
During the 1930s, You Watanabe developed a measure-theoretic formulation of this lattice inversion relation, separating the dependence on lattice span from the choice of lattice origin. In modern notation, if (X) is supported on (a+h\mathbb Z), the mass at (a+kh) is represented by
[ \Pr(X=a+kh)
\frac{h}{2\pi} \int_{-\pi/h}^{\pi/h} e^{-it(a+kh)}\varphi_X(t),dt. ]
The corresponding quasi-periodicity relation is
[ \varphi_X!\left(t+\frac{2\pi}{h}\right)
e^{2\pi ia/h}\varphi_X(t). ]
These formulas identify lattice structure through the behavior of the characteristic function on a bounded frequency interval.
Moments and differentiability
When suitable moments exist, derivatives of the characteristic function at the origin encode them. If (\operatorname{E}[|X|^n]<\infty), then (\varphi_X) is (n) times differentiable and
[ \varphi_X^{(n)}(0)=i^n\operatorname{E}[X^n]. ]
In particular, finite first and second moments give
[ \varphi_X'(0)=i\operatorname{E}[X], \qquad \varphi_X''(0)=-\operatorname{E}[X^2]. ]
A local expansion then yields
[ \varphi_X(t)
1+i\operatorname{E}[X]t -\frac{\operatorname{E}[X^2]}{2}t^2 +o(t^2). ]
Existence of a derivative at zero does not, by itself, imply existence of the corresponding absolute moment without additional conditions. Characteristic functions therefore contain more distributional information than their formal Taylor series.
The logarithm of a characteristic function, when defined consistently near the origin, generates cumulants. For a distribution with sufficiently many moments,
[ \log\varphi_X(t)
\sum_{n=1}^{\infty} \kappa_n\frac{(it)^n}{n!}, ]
where (\kappa_n) is the (n)-th cumulant. Under independence, cumulants add because characteristic functions multiply.
Convergence of distributions
Characteristic functions provide a criterion for convergence in distribution. The Lévy continuity theorem states that if (X_n) has characteristic function (\varphi_n), then
[ X_n \xrightarrow{d} X ]
implies
[ \varphi_n(t)\longrightarrow\varphi_X(t) \quad\text{for every }t\in\mathbb R. ]
Conversely, if (\varphi_n(t)) converges pointwise to a function (\varphi(t)) that is continuous at zero, then (\varphi) is a characteristic function and the corresponding distributions converge weakly to its probability law.
The continuity requirement at zero prevents probability mass from escaping to infinity. This theorem reduces many limit problems to the analysis of pointwise products and local expansions. For independent identically distributed variables with finite variance, that reduction gives a standard characteristic-function proof of the central limit theorem.
If (X_1,X_2,\ldots) are independent and identically distributed with mean (\mu) and variance (\sigma^2>0), then the normalized sum
[ S_n=\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n} ]
has characteristic function
[ \varphi_{S_n}(t)
\left[ e^{-it\mu/(\sigma\sqrt n)} \varphi_X!\left(\frac{t}{\sigma\sqrt n}\right) \right]^n. ]
The second-order expansion at zero shows that this expression converges to (e^{-t^2/2}), the characteristic function of the standard normal distribution.
Representative distributions
For a normal distribution with mean (\mu) and variance (\sigma^2),
[ \varphi_X(t)
\exp!\left(i\mu t-\frac{\sigma^2t^2}{2}\right). ]
For a Poisson distribution with parameter (\lambda),
[ \varphi_X(t)
\exp!\left(\lambda(e^{it}-1)\right). ]
For a Cauchy distribution with location (x_0) and scale (\gamma>0),
[ \varphi_X(t)
\exp!\left(ix_0t-\gamma|t|\right). ]
The Cauchy law illustrates the independence of characteristic functions from moment existence: its characteristic function is defined everywhere, although its ordinary mean and variance do not exist.
The uniform distribution on ([-a,a]), where (a>0), has
[ \varphi_X(t)
\frac{\sin(at)}{at}, ]
with the value at (t=0) assigned by continuity. Its zeros demonstrate that a characteristic function need not remain nonzero away from the origin.
Infinite divisibility
A probability distribution is infinitely divisible when, for every positive integer (n), it is the distribution of a sum of (n) independent identically distributed random variables. In characteristic-function terms, (\varphi) is infinitely divisible when it admits an (n)-th characteristic-function root for every (n).
The Lévy–Khintchine formula characterizes such functions on the real line:
[ \varphi(t)
\exp!\left( i\gamma t -\frac{1}{2}\sigma^2t^2 + \int_{\mathbb R\setminus{0}} \left( e^{itx}-1-itx,\mathbf 1_{{|x|<1}} \right)\nu(dx) \right). ]
Here (\gamma\in\mathbb R), (\sigma^2\geq0), and (\nu) is a Lévy measure satisfying
[ \int_{\mathbb R\setminus{0}} (1\wedge x^2),\nu(dx)<\infty. ]
This representation connects characteristic functions with Lévy processes, whose stationary independent increments correspond to exponential dependence on time in the Fourier domain.
Multivariate form
For a random vector (X\in\mathbb R^d), its characteristic function is
[ \varphi_X(t)
\operatorname{E}!\left[e^{i\langle t,X\rangle}\right], \qquad t\in\mathbb R^d, ]
where (\langle t,X\rangle) denotes the Euclidean inner product. The multivariate function determines the joint distribution, while restriction to a line gives the characteristic function of a linear projection:
[ \varphi_X(su)
\operatorname{E}!\left[e^{is\langle u,X\rangle}\right]. ]
The Cramér–Wold theorem states that a probability law on (\mathbb R^d) is determined by all of its one-dimensional linear projections. This result aligns the multivariate uniqueness theory with the corresponding one-dimensional theory.
Historical development
The use of Fourier integrals in probability emerged from the analytical work of Pierre-Simon Laplace on generating functions and asymptotic distributions. Augustin-Louis Cauchy subsequently employed closely related transforms in his analysis of the distribution now bearing his name.
Paul Lévy established the systematic modern theory of characteristic functions, including their role in weak convergence and stable distributions. Aleksandr Khinchin contributed to the structure theory of infinitely divisible laws, while Harald Cramér developed characteristic-function methods in asymptotic statistics and multivariate distribution theory. Salomon Bochner supplied the general positive-definiteness characterization that places the probabilistic construction within harmonic analysis.