Calculating probability with help of characteristic function

Question

Let $X$ be integer valued with Characteristic Function $\phi$. How to show that

$\ P(|X| = k)= \frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{ikt} \phi(t) dt$

$P(S_n =k) =\frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{ikt} (\phi(t))^n dt $

Answer 1

I don't think that $(1)$ is true.

Take the simplest possible example. Let $P(X=1)=p$ and let $P(X=0)=1-p$. Then, for $k=1$ $$P(|X|=1)=p.$$

The characteristic function of this distribution is $$1-p+pe^{it}.$$

The OP claims that $$p=\frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{it} (1-p+pe^{it}) dt.$$

But $$\frac{1}{2\pi }\int _{-\pi} ^{\pi} e^{it} (1-p+pe^{it}) dt=0.$$

I suspect then that $(2)$ is not true either -- for the same reason. Note that the characteristic function of $S_n$ is $$(1-p+pe^{it})^n.$$

Answer 2

You are using the expression for the probability density function, given the characteristic function. However you are dealing with a discrete distribution, so you must use the expression for the cumulative distribution function from the characteristic function. See the following:

https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29