Concentration of two independent sub-Gaussian random variables Suppose $X$ and $Y$ are independent sub-Gaussian random variables with 0 mean and $\sigma^2$ sub-Gaussian parameter. More specifically, $\mathbb E[\exp(a^T X)]\leq \exp\{\|a\|_2^2\sigma^2/2\}$ for all $a$, and the same holds for $Y$ as well.
I wish to upper bound the tail probability
$$
\Pr\left[|X^T Y|>t\right]
$$
using $\sigma^2$ and dimension $n$ (that is, both $X$ and $Y$ are $d$-dimensional random variables). How can I achieve this?  $X^T Y$ does not seem to be either sub-Gaussian or sub-exponential.

Moment generating functions of subgaussian vector can often be bounded from above by the same moment generating function, with the subgaussian vector replaced by a standard normal. This is equivalent to zhoraster's answer.
Take $\sigma=1$ without loss of generality (otherwise, consider $X/\sigma$ and $Y/\sigma$ and scale back afterwards).
Let $g,h$ be standard normal random vectors, independent of $X,Y$. Then by independence,
\[
E[e^{uX^TY} | X ] = E[e^{u^2\|X\|^2/2} ] = E[e^{u X^\top g}].
\]
Now by the law of total expectation, and a similar argument for $Y$,
\[
E[e^{uX^TY}] \le E[ E[e^{u X^T g}| g] ] = E[E[ e^{u^2\|g\|^2/2} | g]] = E[e^{u g^Th}].
\]
So the problem is reduced to bounding the moment generating function where both random vectors are standard normal.
