Characteristic Function of Random Variable is Well-Defined

Theorem

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 $\phi: \R \to \C$ of $X$ is well-defined.

Proof

Let $t \in \R$.

Recall:

\(\ds \map \phi t\)

\(=\)

\(\ds \expect {e^{i t X} }\)

Definition of Characteristic Function

\(\ds \)

\(=\)

\(\ds \int e^{i t X} \rd \Pr\)

Definition of Expectation

Thus we need to show that the last integral exists.

By Modulus of Exponential of Imaginary Number is One:

$\cmod {e^{i t X } } = 1$

since $t \map X \omega \in \R$ for all $\omega \in \Omega$.

In particular:

$\ds \int \cmod {e^{i t X} } \rd \Pr = \int 1 \rd \Pr = 1$

This article, or a section of it, needs explaining. In particular: No seriously, I mean that, a link to the definition of $\d \Pr$ which is understandable in the context of the general measure $\mu$, which in itself is not a trivial concept to grab, so at least thrown a bone to the poor reader of the page, yeah? I gave it above as Definition of Expectation. Or feel free to rename $\Pr$ to $\mu$, if it is somehow helpful. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Thus, in view of Characterization of Integrable Functions:

$\ds \int e^{i t X} \rd \Pr$

exists.

$\blacksquare$