I am studying the relationship / bijection between characteristic functions and CDFs.
In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function as:
$$F(x) =\frac{1}{2}+\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{e^{iux}\phi(-u)-e^{-iux}\phi(u)}{iu}du$$
Could anyone give me some insights on how to prove this result?
Any hint will be greatly appreciated
First of all, look to the integral between $-T$ and $T$, the last thing you will do is to let $T \rightarrow \infty$.
Ok? Now substitute $\phi(u)$ by his integral expression. You will have a double integral.
Then you will have to convince yourself that you can change the order of integration (Fubini's Theorem could help)...
After this you can think about pass the limit in $T$ inside one integral (how about Dominated convergence theorem?)
This is just an idea that I know works. But this theorem isn't too simple to proof (my opinion) so you can see the proof in any good book of probability. You can try Shiryaev's or Durrett's ... both are good books.