Characteristic function (${\phi }_X(t)$)

The characteristic function of a real random variable $X$ isIt determines the probability distribution of $X$.

For a real random variable $Y$ and a sigma-algebra $\mathcal{G}$, the conditional characteristic function is $\mathbb{E}[e^{i\theta Y}\mid \mathcal{G}]$. If it equals the nonrandom characteristic function of a probability law for every $\theta$, then $Y$ has that law and is independent of $\mathcal{G}$. To verify the independence assertion, multiply the identity by any bounded $\mathcal{G}$-measurable test variable and use the uniqueness theorem for characteristic functions on the resulting finite measures. Equality for rational $\theta$ extends by continuity, so simultaneous null sets can be arranged when needed.

For a sample $X_1,\dots ,X_n$, ${\phi }_n(u)=n^{-1}{\sum }_j{e}^{iu{X}_j}$ is the characteristic function of the sample's empirical measure. For independent and identically distributed random variables, $\mathbb{E}{\phi }_n(u)=\phi (u)$ and ${\mathrm{Var}}_{\mathbb{C}}{\phi }_n(u)=(1-|\phi (u){|}^2)/n$, without any moment condition on the observations.

The centered process ${\mathcal{C}}_n(u)=\sqrt{n}({\phi }_n(u)-\phi (u))$ describes fluctuations of the empirical characteristic function. At a fixed frequency, its real and imaginary parts satisfy a multivariate central limit theorem. A process-level limit requires further control over the frequency index.

Every characteristic function is uniformly continuous and satisfies $\phi (0)=1$ and $|\phi (t)|\le 1$. Continuity follows from the dominated convergence theorem applied to $e^{itX}$.

If $|{\phi }_X(u)|=1$, then $e^{iuX}$ is almost surely constant. Indeed,Hence $uX\in \theta +2\pi \mathbb{Z}$ almost surely for some real $\theta$.

If $u/v$ is irrational and $|{\phi }_X(u)|=|{\phi }_X(v)|=1$, then $X$ is almost surely constant. For independent copies $X,X^{\prime }$, both $u(X-X^{\prime })$ and $v(X-X^{\prime })$ lie in $2\pi \mathbb{Z}$ almost surely; irrationality makes the two corresponding lattices intersect only at zero, so $X=X^{\prime }$ almost surely. Two independent identically distributed variables can agree almost surely only when their common law is a point mass.

For an ${\mathbb{R}}^d$-valued random vector, the characteristic function is ${\phi }_X(u)=\mathbb{E}[e^{i⟨u,X⟩}]$. It determines the joint distribution and satisfies ${\phi }_{AX}(u)={\phi }_X(A^{T}u)$ for every deterministic matrix $A$.

If $X\sim N(0,{\sigma }^2{I}_d)$, then

A random vector $X\in {\mathbb{R}}^d$ is orthogonally invariant when $AX$ has the same distribution as $X$ for every $A\in O(d)$. Its characteristic function is radial and real: ${\phi }_X(u)=f(\Vert u{\Vert }^2)$ for a continuous function $f:[0,\infty )\to \mathbb{R}$.

If an orthogonally invariant random vector in dimension at least two has independent coordinates, then it is centered isotropic Gaussian, possibly degenerate at zero. Independence and rotational invariance givewhile continuity and $f(0)=1$ imply $f(r)=e^{-\alpha r}$ for some $\alpha \ge 0$. Thus $X\sim N(0,2\alpha I_d)$.

Two finite Borel measures on $\mathbb{R}$ with the same characteristic function are equal. One proof convolves both measures with centered Gaussians, applies Fourier inversion to the smooth densities, and then lets the Gaussian variance tend to zero.

Let $f$ be compactly supported and continuous, let $g,\hat{g}\in L^1(\mathbb{R})$, and use $\hat{g}(t)=\int g(x)e^{itx}dx$. If $g(0)\ne 0$, thenThis follows by Fubini's theorem, a change of variables, Fourier inversion theorem, and dominated convergence theorem. It gives a direct proof of the uniqueness theorem for characteristic functions because the right side depends only on ${\phi }_Z$.

Independence makes the characteristic function of a sum equal the product of the individual characteristic functions.

If the characteristic functions ${\phi }_n$ of probability distributions ${\mu }_n$ converge pointwise to a function $\phi$ that is continuous at zero, then $\phi$ is a characteristic function and ${\mu }_n$ converges in distribution to its distribution. Conversely, convergence in distribution implies pointwise convergence of the characteristic functions.

If independent and identically distributed random variables have characteristic function $\phi (t)=1+iat+o(|t|)$ at zero, then their sample mean has convergence in distribution to $a$: its characteristic function is $\phi (t/n{)}^n\to e^{iat}$. This is also convergence in probability by convergence in distribution to a constant implies convergence in probability.