This is my first question in this site. First of all, thank you for reading. The question is:
For some positive integer $s$, let $X_{1}, X_{2} \ldots $ be independent random variables such that $$X_{i}\sim\frac{1}{i!}\chi^{2}(s^i).$$ Also, let $S=\sum^{\infty }_{i=1}X_{i}-e^{s}$.
I can show that the (absolute) first and second moments of $S$ exist. However, I am not sure if (absolute) $p$ moments of $S$ exist for any positive integer $p$. Can somebody please answer this question?