Characteristic functions convert convolution and weak convergence into pointwise analytic statements. They are the main Fourier tool of classical limit theory.
Fourier transform of a law
The characteristic function is always defined and always bounded.
Definition. For a real random variable \(X\), its characteristic function is \[ \varphi_X(t)=\mathbb E e^{itX},\qquad t\in\mathbb R. \] For \(X\in\mathbb R^d\), use \(\varphi_X(t)=\mathbb E e^{i\langle t,X\rangle}\).
Proposition. Every characteristic function satisfies \[ \varphi_X(0)=1,\qquad |\varphi_X(t)|\le1,\qquad \varphi_X(-t)=\overline{\varphi_X(t)}, \] and is uniformly continuous. It is positive definite: \[ \sum_{j,k=1}^nc_j\overline{c_k}\, \varphi_X(t_j-t_k)\ge0. \]
Proof. The first identities are immediate. Uniform continuity follows from \[ \sup_t|\varphi_X(t+h)-\varphi_X(t)| \le\mathbb E|e^{ihX}-1|\longrightarrow0 \] by dominated convergence. Finally, \[ \sum_{j,k}c_j\overline{c_k}\varphi_X(t_j-t_k) =\mathbb E\left|\sum_jc_je^{it_jX}\right|^2\ge0. \]
Uniqueness and Levy continuity
A law is determined by its characteristic function, and pointwise convergence at a continuous limit identifies weak convergence.
Theorem. If \(\varphi_X=\varphi_Y\), then \(X\stackrel d=Y\). If \(\varphi_n(t)\to\varphi(t)\) pointwise and \(\varphi\) is continuous at \(0\), then \(\varphi\) is the characteristic function of a probability law \(\mu\), and the corresponding laws \(\mu_n\Rightarrow\mu\).
Proof. Let \(\gamma_\varepsilon\) be \(N(0,\varepsilon)\). If \(\varphi_\mu=\varphi_\nu\), then \[ \widehat{\mu*\gamma_\varepsilon} =\varphi_\mu(t)e^{-\varepsilon t^2/2} =\varphi_\nu(t)e^{-\varepsilon t^2/2} =\widehat{\nu*\gamma_\varepsilon}. \] The product is integrable, so Fourier inversion gives equal smooth densities. As \(\varepsilon\downarrow0\), Gaussian convolution converges weakly to the original measure, proving \(\mu=\nu\).
For Levy continuity, continuity of \(\varphi\) at zero yields tightness through a smoothing inequality that bounds \[ \mu_n(|x|>R) \] by an average of \(1-\operatorname{Re}\varphi_n(t)\) over \(|t|\lesssim1/R\). Every subsequence therefore has a weakly convergent further subsequence. Weak convergence passes characteristic functions to the limit, so every subsequential law has characteristic function \(\varphi\). Uniqueness makes all subsequential limits equal, proving full weak convergence. The limit is automatically a characteristic function.
Sums and moments
Independence changes sums into products.
Proposition. If \(X\) and \(Y\) are independent, then \(\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)\). If \(\mathbb E|X|^k\lt\infty\), then \(\varphi_X\) has derivatives up to order \(k\) at zero with \(\varphi_X^{(k)}(0)=i^k\mathbb E X^k\).
Proof. Independence gives \[ \mathbb E[e^{itX}e^{itY}] =\mathbb Ee^{itX}\,\mathbb Ee^{itY}. \] For derivatives, \[ \frac{\mathrm{d}^j}{\mathrm{d}t^j}e^{itX} =(iX)^je^{itX}, \] whose absolute value is bounded by \(|X|^j\le1+|X|^k\) for \(j\le k\). Dominated differentiation yields the formula.
Proposition (second-order expansion). If \(\mathbb EX=0\) and \(\mathbb EX^2=\sigma^2<\infty\), then \[ \varphi_X(t)=1-\frac{\sigma^2t^2}{2}+o(t^2) \qquad(t\to0). \]
Proof. Use \[ e^{iu}=1+iu-\frac{u^2}{2}+u^2r(u), \qquad r(u)\to0. \] After substituting \(u=tX\), the remainder divided by \(t^2\) is \(X^2r(tX)\). A bounded version of \(r\) and dominated convergence with \(X^2\) show its expectation tends to zero.
Canonical characteristic functions. \[ \begin{array}{c|c} \text{law}&\varphi(t)\\ \hline N(\mu,\sigma^2)&\exp(i\mu t-\sigma^2t^2/2)\\ \operatorname{Poisson}(\lambda)&\exp(\lambda(e^{it}-1))\\ \operatorname{Bernoulli}(p)&1-p+pe^{it}\\ \operatorname{Cauchy}(0,\gamma)&e^{-\gamma|t|} \end{array} \] The Cauchy characteristic function is not differentiable at \(0\), matching the failure of its first absolute moment.
Remark. Characteristic functions are especially robust for triangular arrays because no densities or smoothness of distributions are required.