Suppose we have $n$ observations $(x_i,y_i)\in\mathbb{R}^{p+1}$. Suppose the underlying model is $y = \eta^{*T} x + \varepsilon$, and $x, \varepsilon$ are sub-Gaussian with parameter $\sigma_x, \sigma_{\epsilon}$, respectively.
Under regularity assumptions, it is well known that the OLS estimator of the coefficient is $\hat \eta = (X^TX)^{-1}X^TY$, where $X\in\mathbb{R}^{n x p}$ and $Y\in \mathbb{R}^{n}$, we have $$\hat \eta = (X^TX)^{-1}X^TY = (X^TX)^{-1}X^T (X\eta^* + \mathcal{E}) = (X^TX)^{-1}X^T X\eta^* + (X^TX)^{-1}X^T \mathcal{E} = \eta^*+ (X^TX)^{-1}X^T \mathcal{E} $$
I want to know if $\hat \eta$ is a sub-Gaussian (or sub-Exponential)? If so, what is the sub-Gaussian (or sub-Exponential) parameter?
Since $\eta^*$ is non-random, I just need to focus on $(X^TX)^{-1}X^T \mathcal{E}$. Recall that $X$ is a $n$ x $p$ sub-Gaussian random matrix, $\mathcal{E}$ is a $n$ x $1$ sub-Gaussian random vector, is there any existing theorem talks about such case?