My question is that how much information we can get form integer moments of a complex random variable?

Let $\mathcal{Z}$ be a complex value random variable, and assume that we can compute $$\int \mathcal{Z}^k d\mu,$$ For $k \in \mathbb{N}$ and $\mu$ be a measure.

I also am looking for an example of a complex random variable, that its moments positively approaches zero with

$$ 0< \int \mathcal{Z}^k d\mu \asymp \frac{1}{k!}.$$

If moments satisfy the above, can we occlude that $|\mathcal{Z}|$ is bounded almost surely?

One simple example: let $$ \mathcal{Z}= \sum_{m \in A} e^{imt} + \tfrac{1}{2},$$

where $A \subset \mathbb{Z}$ and $t$ is uniformly distributed in $(0, 2\pi].$ Then k-th moment of $\mathcal{Z}$ is 2^{-k}. Therefore moments decay but they fall short of decaying like $1/k!$.