Definition:Characteristic Function of Random Variable

This page is about Characteristic Function in the context of Probability Theory. For other uses, see Characteristic Function.

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

• Results about characteristic functions of random variables can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): characteristic function
• 2002: George C. Casella and Roger L. Berger: Statistical Inference (2nd ed.): $2.6.2$: Other Generating Functions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): characteristic function: 1.