How to find discrete distribution function from characteristic function?
I came across the problem in my assignment and I don't know how to calculate it. I know how to calculate continuous distribution function from characteristic functions.
Here is my problem:
For $n\ge 0$, let $a_n$ be non-negative real numbers such that $\sum_{n=1}^\infty a_n=1$. Show that the function $$\varphi(t):=\sum_{n=1}^\infty a_n \cos(nt)$$ is the characteristic function of a distribution function. What is the corresponding distribution?
My attempt is to make $$\sum_{n=1}^\infty a_n \cos(nt) = \sum_{n=1}^\infty a_n \cdot (\exp(int)+\exp(-int))/2 = \sum_{n=1}^\infty \exp(int)P_x(n)$$ However, I don't know the following steps to calculate it.
Take $p_n =p_{-n}=\frac {a_n} 2$ for $n=1,2,...$ and $p_0=a_0$. ($\phi$ is the characteristic function of a random variable which takes the values $n$ and $-n$ with probability $\frac {a_n} 2$ and $0$ with probability $a_0$).