Characteristic Function
# Definition
Suppose $X$ is a Random Variable on Probability Space $(\Omega, \mathcal{M}, \mathbb{P})$. The Characteristic Function of $X$ is a function $\phi_{X}: \mathbb{R} \to \mathbb{C}$ defined as
$$\phi_{X}(t) = \mathbb{E}(e^{itX})$$
# Elementary Properties
- $X$ is a Random Variable. $\phi_{X}(t)$ exists from all $t \in \mathbb{R}$. This follows because $|e^{itX}| = 1$. Thus, $\mathbb{E}(|e^{itX}|) = \mathbb{E}(1) = 1$ and $e^{itX} \in L^{1}(\Omega)$.
- $X, Y$ are Independent Random Variables. Then $$\begin{align*} \phi_{X+Y}(t) &= \mathbb{E}(e^{it(X+Y)})\\ &=\mathbb{E}(e^{itX} e^{itY})\\ &=\mathbb{E}(e^{itX}) \mathbb{E}(e^{itY}) & \text{(Independence)}\\ &=\phi_{X}(t) \phi_{Y}(t) \end{align*}$$
- $X$ is a Random Variable, $a, b \in \mathbb{R}$. Then $$\begin{align*} \phi_{aX+b}(t) &= \mathbb{E}(e^{it(aX+b)})\\\ &=\mathbb{E}(e^{itaX} e^{itb})\\ &=e^{itb}\mathbb{E}(e^{i(ta)X})\\ &=e^{itb} \phi_{X}(ta) \end{align*}$$
- $X$ is a Random Variable. Applying Euler’s Formula
# Passing the Expectation into the Taylor Expansion of the Exponential
$X$ is a Random Variable. We would like to understand how we can apply the Taylor Expansion of the Exponential to our definition of a Characteristic Function. $$\begin{align*} \phi_{X}(t) &= \mathbb{E}(e^{itX})\\ &=\mathbb{E}\left(1 + \sum\limits_{k=1}^{\infty} \frac{(itX)^{k}}{k!}\right) \end{align*}$$ where the last equality makes sense because the Taylor Expansion of the Exponential has unbounded Radius of Convergence. However, it does not necessarily follow that we can “apply” Linearity of Expectation to the sum. For example, the example from Expectation does not always exist has characteristic function value for each $t \in \mathbb{R}$ by property (1), but passing the expectation through the taylor expansion has no clear meaning.
However, with some more assumptions we can say more:
# Existence of Moment Generating Function
Suppose $m_{X}(t) = \mathbb{E}(e^{tX})$ exists for some $t > 0$. Observe that $$\begin{align*} e^{|tX|} &= 1 + \sum\limits_{k=1}^{\infty} \frac{|tX|^{k}}{k!} \\ &= 1 + \sum\limits_{k=1}^{\infty} \frac{|itX|^{k}}{k!}\\ &= 1 + \sum\limits_{k=1}^{n} \frac{|itX|^{k}}{k!}\\ &\geq \bigg| 1 + \sum\limits_{k=1}^{n} \frac{(itX)^{k}}{k!} \bigg| \\ \end{align*}$$ Existence of Moment Generating Functions imply Existence of Moments, so TODO.
$$\begin{align*} &\mathbb{E}(e^{tX} 1_{X \geq 0}) \leq \mathbb{E}(e^{tX})\\ &\mathbb{E}(e^{-tX} 1_{X < 0}) \leq \mathbb{E}(e^{-tX})\\ &\mathbb{E}(e^{tX} 1_{X \geq 0}) + \mathbb{E}(e^{-tX} 1_{X < 0}) = \mathbb{E}(e^{t|X|}) \leq m_{X}(t) + m_{X}(-t) \end{align*}$$