I have given the characteristic function $$\mathcal F(p)(x)=\frac{\exp(-|x|^s)}{(1 + x^2)^5}$$ where $s=0.5, s=1, s=1.5$ or $s=2$.
Now I want to get sample values of the distribution belonging to the density $p$.
Since the inverse Fourier transform does not exist in closed form one way to get samples would be using numerical methods. A well-known method is "Inverse transform sampling" but for this I would need the inverse CDF and it is a long way to get there (CF->PDF->CDF->InvCDF).
Does anyone know a better algorithm to generate a sample? My paper refers to the properties of $\alpha$-stable random variables.