# Characteristic functions (aka Fourier Transforms)

The characteristic function (or Fourier Transform) of a random variable $$X$$ is defined as
\begin{align*}
\psi(t)= \mathbf E \exp( i t X)
\end{align*}
for all $$t \in \mathbf R$$.

It is a basic fact that the characteristic function of a random variable uniquely determine the distribution of a random variable. Furthermore, the following convergence theorem is classical theorem from probability theory.

Theorem: Let $$X_n$$ be a sequence of real-valued random variable with characteristic $$\psi_n$$ be the associated characteristic functions. Assume that there exists a function $$\psi$$ so that for each $$t \in \mathbf R$$
$\lim_{n \rightarrow \infty} \psi_n(t) = \psi(t)$
If $$\psi$$ is continuous at zero then there exists a random variable $$X$$ so that the distribution of $$X_n$$ converges to the distribution of $$X$$. Furthermore the characteristic function of $$X$$ is $$\psi$$.