The characteristic function is the Fourier transform of the probability density.

Note that $\exp (i{\mathbf{t}}^{\mathrm{\top }}\mathbf{x})=\cos ({\mathbf{t}}^{\mathrm{\top }}\mathbf{x})+i\sin ({\mathbf{t}}^{\mathrm{\top }}\mathbf{x})$; the characteristic function is thus bounded and continuous, so by the Portmanteau theorem, if ${\mathbf{x}}_n\stackrel{\text{d}}{⟶}\mathbf{x}$ then the characteristic functions will also converge. This is actually an if and only if statement, known as the continuity theorem and formally stated below.

Important properties of characteristic functions

For all of the properties below, $x$ and $y$ are random variables with characteristic functions ${\phi }_{\mathbf{x}}$ and ${\phi }_{\mathbf{y}}$, $b$ and $\mathbf{c}$ are constants, and $\mathit{\mu }=\mathbb{E}\mathbf{X}$. To keep things consistent with the numbering in the course notes, I will refrain from numbering the first three properties.

Existence and uniqueness

(i) For any random vector $\mathbf{x}$, $\phi (\mathbf{t})$ exists and is continuous for all $\mathbf{t}\in {\mathbb{R}}^d$.

(ii) Two random vectors $\mathbf{x}$ and $\mathbf{y}$ have the same distribution if and only if ${\phi }_{\mathbf{x}}(\mathbf{t})={\phi }_{\mathbf{y}}(\mathbf{t})$.

Continuity

Other properties

(1) $\phi (\mathbf{0})=1$ and $|\phi (\mathbf{t})|\le 1$ for all $\mathbf{t}$

(2) ${\phi }_{\mathbf{x}/b}(\mathbf{t})={\phi }_{\mathbf{x}}(\mathbf{t}/b)$ for $b\ne 0$

(3) ${\phi }_{\mathbf{x}+\mathbf{c}}(\mathbf{t})=\exp (i{\mathbf{t}}^{\mathrm{\top }}\mathbf{c}){\phi }_{\mathbf{x}}(\mathbf{t})$

(4) ${\phi }_{\mathbf{x}+\mathbf{y}}(\mathbf{t})={\phi }_{\mathbf{x}}(\mathbf{t}){\phi }_{\mathbf{y}}(\mathbf{t})$ if $\mathbf{x}\perp \perp \mathbf{y}$

(5) $\mathrm{\nabla }{\phi }_{\mathbf{x}}(\mathbf{t})$ exists, is continuous, and $\mathrm{\nabla }{\phi }_{\mathbf{x}}(\mathbf{0})=i\mathit{\mu }$ if $\mathbb{E}\Vert \mathbf{x}\Vert <\mathrm{\infty }$

(6) ${\mathrm{\nabla }}^2{\phi }_{\mathbf{x}}(\mathbf{t})$ exists, is continuous, and ${\mathrm{\nabla }}^2{\phi }_{\mathbf{x}}(\mathbf{0})=-\mathbb{E}(\mathbf{x}{\mathbf{x}}^{\mathrm{\top }})$ if $\mathbb{E}\Vert \mathbf{x}{\Vert }^2<\mathrm{\infty }$

(7) ${\phi }_{\mathbf{x}}(\mathbf{t})=\exp (i{\mathbf{t}}^{\mathrm{\top }}\mathbf{c})$ if $\mathbf{x}=\mathbf{c}$ with probability 1

(8) ${\phi }_{\mathbf{x}}(\mathbf{t})=\exp (i{\mathbf{t}}^{\mathrm{\top }}\mathit{\mu }-\frac{1}{2}{\mathbf{t}}^{\mathrm{\top }}\mathbf{\Sigma }\mathbf{t})$ if $\mathbf{x}\sim \text{N}(\mathit{\mu },\mathbf{\Sigma })$