How do you invert a characteristic function, when integral does not converge?

Question

I need to find the probability density of some distribution with characteristic function given by:

$$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$

I know the formula for inverting a characteristic function is:

$$f_X(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \phi(\omega) e^{-i\omega x} \mathop{d\omega}$$

But obviously putting this function inside the formula, will make the integral diverge. So my question is how does one invert a characteristic function, when this integral diverges? Or is this supposed to always converge and there is something wrong with my characteristic function?

Answer 1

The inversion formula you cite is restricted to integrable characteristic functions. The case in your question is $\varphi_X(\omega)=\sum\limits_{k=1}^np_k\mathrm e^{\mathrm i \omega a_k}$ with $p_k\gt0$ and $\sum\limits_{k=1}^np_k=1$, which is never integrable.

Assume that $n=1$, that is, that $\varphi_X(\omega)=\mathrm e^{\mathrm i \omega a}$. Can you identify the distribution of $X$ in this case? Hint: there is no density. Then the general case might be straightforward.