Let's say we have the usual linear model $$Y = \beta^{T}X + \varepsilon,$$
where $Y \in \mathbb{R}, X \in \mathbb{R}^p, \beta \in \mathbb{R}^p$ and $\varepsilon \in \mathbb{R}, \varepsilon \sim N(0,\sigma^2)$. Furthermore $Y \sim N(0,\sigma_Y^2)$ and $X \sim N(0,\Sigma)$. Suppose that we have centered data without intercept. The OLS estimator is $$\hat{\beta} = \left(X^T X \right)^{-1}X^T Y = \beta + \left(X^T X \right)^{-1}X^T \varepsilon.$$
We already know that CLT says (with some assumptions) $$\sqrt{N} \left(\hat{\beta} - \beta \right) \sim N\left(0, \sigma^2 \left(N^{-1}X^T X\right)\right)$$ if $N \rightarrow \infty.$
Now I wonder, let's say $X_N$ is a new independent observation of $X$. So $Y_N = \beta^{T}X_N + \varepsilon_N$ and $\widehat{Y}_N = \hat{\beta}^{T}X_N$. What can I say about the distribution of $\sqrt{N} \left(\hat{\beta}^TX_N - \beta^T X_N \right)$?