Properties of a Characteristic Function

Topics: Probability - Characteristic Function

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(theorem)

Let $X$ be any random variable. Then, with respect to its characteristic function ${\phi }_X(t)$. we have that:

1. The characteristic function of any random variable depends only on its distribution.

2. ${\phi }_X(0)=\mathbb{E}[e^0]=1$

3. The characteristic function takes values within the unit circle (i.e. $|{\phi }_X(t)|\le 1\forall t\in \mathbb{R}$

4. Let $Y$ be another random variable independent from $X$. Then ${\phi }_{X+Y}(t)={\phi }_X(t){\phi }_Y(t)$

5. If $X$ and $Y$ are random variables such that ${\phi }_X(t)={\phi }_Y(t)\forall t\in \mathbb{R}$, then $X$ and $Y$ have the same distribution.

6. The characteristic function is a continuous function.

With Respect to Moments §

(theorem)

Let $X$ be a random variable with an $n$th finite moment. Then:

1. The $n$th derivative of ${\phi }_X(t)$ exists and is continuous.

\left. \frac{d^n \varphi_{X}(t)}{dt^n} \right|{t=0} &= i^n \mathbb{E}[ X^n ] \[1em] &\implies \[1em] \mathbb{E}[ X^n ] &= \frac{1}{i^n} \left. \frac{d^n \varphi{X}(t)}{dt^n} \right|_{t=0} \end{align*}

# With Respect to Linear Functions ***(theorem)*** Let $a, b \in \mathbb{R}$ and $Y = aX+b$ with $X$ a [[Random Variable|random variable]]. Then:

\varphi_{Y}(t) = e^{ibt} \varphi_{X}(at)

# With Respect to Independent Random Variables ***(theorem)*** Let $X_{1}, \dots, X_{n}$ be [[Event Independence|independent]] random variables. Then:

\varphi_{\sum \limits_{i=1}^{n} x_{i}}(t) = \prod_{k=1}^{n} \varphi_{X_k}(t)

> [!warning]- The reciprocal is not true > > Note that the reciprocal is **not** true. That is: > > $$ > \begin{align*} > \varphi_{X_1 + X_2}(t) &= \varphi_{X_1}(t) \varphi_{X_2}(t) \\[1em] > &\centernot \implies \\[1em] > X_1 \text{ and } X_2 &\text{ are independent} > \end{align*} > $$