I am struggling with this concept (self-study). Could someone show me how to explicitly apply the inversion formula for these examples? I am working through about 15 examples, but these 3 seemed sufficiently different to help me do the rest.
$\phi_1(t)=(1-|t|)_+$
$\phi_2(t)=\sum_{n=-\infty}^{\infty}\phi_1(t+2n\pi)$
$\phi_3(t)=(1-\frac{t^2}{2})e^{-t^2/2}$
Work/Thoughts
The only reference to the inversion formula that I have found is the following theorem:
Assumptions:
1-$\phi$ is a characteristic function of a given probability distribution $F$
2- $F$ has continuity points $a, b$ with $a.
$F(b)-F(a)=\lim_{n\to\infty}\frac1{2\pi}\int_{-\infty}^{\infty}\frac{e^{-ita}-e^{-itb}}{it}\phi(t)e^{-t^2/n}dt$
And I believe this simplifies to:
$\lim_{n\to\infty}\frac1{2\pi}\int_{-n}^{n}\frac{e^{-ita}-e^{-itb}}{it}\phi(t)$
From here I am not sure what to do. Thanks for any help.
more thoughts
I have read about two kinds of invertible CFs- those that are integrable, and those that are periodic. $\phi_2(t)$ is obviously of the periodic nature.
I also understand the following properties about characteristic functions:
If $F$ and $G$ are probability distributions and $G$ is absolutely continuous, then $F*G$ has density $\int_{-\infty}^{\infty}g(u-x)F(dx)$
This this helpful at all, perhaps for number 3?