Characteristic functions (aka Fourier Transforms)

The characteristic function (or Fourier Transform) of a random variable \(X\) is defined as
\psi(t)= \mathbf E \exp( i t X)
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 \(\psi\).