I asked this question at math.stackexchange but got no satisfactory answers. Thus I figure it might be potentially helpful to ask the question here.

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 $n$-dimensional random variables). How can I achieve this? $X^T Y$ does not seem to be either sub-Gaussian or sub-exponential.