Note in advance, due to my question previously being marked as a duplicate, that the question I ask here is concerned with the relationship between $(\mathbf{X}^T \mathbf{X})^{-1}$ and $\text{Cov}[\mathbf{X},\mathbf{X}]$, not the relationship between $(\mathbf{X}^T \mathbf{X})^{-1}$ and $\text{Cov}[\mathbf{X},\mathbf{y}]$ which is covered here.
The matrix formulation of multiple regression for $n$ observations is $$ \mathbf{Y} = \mathbf{X}^T \beta + \varepsilon, $$ where the error $\varepsilon$ has finite variance $\sigma^2$. Let $\mathbf{b}$ be the estimated coefficients found when we solve the multiple regression problem with least squares.
In Theorem 4.3 of the book Econometric Analysis by William H. Greene it says that asymptotically $\mathbf{b}$ is distributed as $$ \mathbf{b} = \mathcal{N}\bigg(\beta,\frac{\sigma^2}{n}Q^{-1}\bigg), $$ where $Q$ is defined in equation 4.19 as \begin{align*} Q := \text{plim}_{n \to \infty} \frac{\mathbf{X}^T \mathbf{X}}{n}. \end{align*} which is a positive definite matrix. Suppose we write the covariance matrix of $\mathbf{X}$ as $$ \text{Cov}[\mathbf{X},\mathbf{X}] = \begin{bmatrix} \sigma_{X_1 X_1} & \sigma_{X_1 X_2} & \dots \\ \sigma_{X_2 X_1} & \sigma_{X_2 X_2} & \dots \\ \vdots & & \ddots \\ \sigma_{X_n X_1} & \dots & \dots & \sigma_{X_n X_n} \end{bmatrix}, $$ where $$ \sigma_{X_i X_j} = E[(X_i-E[X_i])(X_j-E[X_j])], $$
I want to know how the matrix $(\mathbf{X}^T \mathbf{X})^{-1}$ relates to the covariance matrix $\text{Cov}[\mathbf{X},\mathbf{X}]$.
For example, if we modify the elements of $\mathbf{X}$ to make them more collinear, the off-diagonal elements of $\text{Cov}[\mathbf{X},\mathbf{X}]$ will increase in magnitude and $(\mathbf{X}^T \mathbf{X})^{-1}$ will become 'harder' to invert. But I am looking for a precise mathematical expression that relates the two matrices.
Can $(\mathbf{X}^T \mathbf{X})^{-1}$ be expressed in terms of $\text{Cov}[\mathbf{X},\mathbf{X}]$ somehow? E.g., can the elements of $(\mathbf{X}^T \mathbf{X})^{-1}$ be written in terms of the elements of $\text{Cov}[\mathbf{X},\mathbf{X}]$?
