Definition:Characteristic Function of Random Variable

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

The characteristic function of $X$ is the mapping $\phi: \R \to \C$ defined by:


 * $\map \phi t = \expect {e^{i t X} }$

where:
 * $i$ is the imaginary unit
 * $\expect \cdot$ denotes expectation.

Also see

 * Characteristic Function of Random Variable is Well-Defined
 * Lévy's Inversion Formula