I have a characterisitic function of the form $e^{(-\alpha t + t^2)}\sum_{j=0}^\infty C_j t^j $, where $\alpha, C_j$'s are complex constants. I am trying to find the probability distribution for this characteristic function.
Had the characteristic function just been of the form $e^{(-\alpha t + t^2)} $, the integration would have been simply
\begin{align} \int_{-\infty}^\infty e^{(-\alpha t + t^2)} e^{-itx} dx &= \int_{-\infty}^\infty e^{-(t^2 + (\alpha + ix)t)} dx \\ &= e^{(\frac{(\alpha+ix)}{2})^2} \int_{-\infty}^\infty e^{-(t+\frac{(\alpha+ix)}{2})^2} dx \end{align}
Using clues from this answer, I think the integral evaluates to some constant. However, having this extra term $\sum_{j=0}^\infty C_j t^j$ troubles me with the integration as I cannot proceed with the above approach.
Can you please help me out. Thank you!
