# Concentration of two independent sub-Gaussian random variables

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.

• The norms $\|X\|$ and $\|Y\|$ are sub-Gaussian with sub-Gaussian parameter $O(d \sigma^2)$. That should be enough to extract bounds on the moments $\mathbb E |X^\top Y|^p$ to obtain tail bounds. (I believe the resulting computation shows that $X^\top Y$ is indeed $d \sigma^2$ sub-Exponential.) – Jon Aug 25 '17 at 15:55