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?
Best Answer
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.