The question is given as follow:
Let $N$ have a Poisson distribution with mean $\lambda$. $X_i$ is Cauchy distribution with mode 0 and and scaling parameter $1$.
Find the characteristic function for $Z = \sum\limits_{i=1}^{N} X_i$. Using this representation, show that $Z$ is heavy tailed.
For the first part, the characteristic function of $X_i$ is $\exp(-|t|)$. The probability generating function of Poisson distribution is $Es^N = e^{\lambda(s-1)}$ Hence
$$E e^{itZ} = e^{\lambda(e^{-i|t|}-1)}$$
It is unclear to me how this means that $Z$ is heavy tailed.
By heavy tailed, we mean that $E e^{\tilde{t}Z}$ does not exist for any $t>0$. so does that mean we set $t = -i\tilde{t}$ in the function for the characteristic function and realise this is not a real function. (so, does that mean if the mgf exists, it must be be an analytic continuation of the characteristic function?)
