I have some question about the difference between two iid Poisson random variables. Given $X,Y \sim Poi(\chi)$ are two iid random variables. We know that $Z=X-Y$ follows from a Skellam distribution. Then, I have a question about the concentration of $Z$. Is there any way we can show $Z$ is sub-gaussian or disprove it? Does anyone have any idea or hint?
Thank you in advance!
P.S. The definition of being sub-gaussian follows from the textbook by
Vershynin, R., 2018. High-dimensional probability: An introduction with applications in data science (Vol. 47). Cambridge university press.