We have two independent Gaussian vectors: $X \sim N(0,I_k)$, $Y \sim N(0,I_r)$ independent of $X$. I'm looking for a tail bound on the product of their square norms.
$Pr_{X,Y}[\|X\|^2\cdot \|Y\|^2 > t]\le ?$
when $t$ is larger than some threshold.
Even a ploynomially decaying bound would be sufficient.