Component-wise product of sub-Gaussians

Question

Let $X_1$ and $X_2$ be two $0$-sub-Gaussians in $\mathbb{R}^n$ of parameter $r$ (both).

Let $Z:=X_1 \odot X_2$ denotes the component-wise multiplication.

What would be the distribution of $Z$? And if $Z$ is a subgaussian, what would be its parameter?