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 $1/k!$.



Moments of complex random variables
This is to rewrite the excellent example by Mateusz Wasilewski in a more conventional form.


Let $Z:=XU$, where $X$ and $U$ are independent random variables (r.v.'s); $P(X>0)=1$; $X$ is unbounded; $EX^k<\infty$ for all natural $k$; $U=e^{iT}$; $T$ is a r.v. with values in the interval $[0,2\pi)$ and pdf $p$ given by the formula
$$p(t)=\frac1{2\pi}\,\Big(1+2\sum\_{n=1}^\infty a\_n\cos nt\Big)$$
for $t\in[0,2\pi)$;
$$0<a\_n\sim\frac1{n!\,EX^n};$$
and $2\sum\_{n=1}^\infty a\_n<1$ (so that $p>0$).


Then for all natural $k$
$$EU^k=Ee^{ikT}=\int\_0^{2\pi}e^{ikt}p(t)\,dt=a\_k$$
and hence
$$EZ^k=EX^k\,EU^k=EX^k\,a\_k\sim\frac1{k!}$$
and $EZ^k>0$, whereas the r.v. $|Z|=X$ is unbounded.


