Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{i,j})$, In this case, the matrix $G= XX^{T}$ follows Wishart distribution and its general inverse $G^{-1}$ follows inverse non-singular Wishart distribution with entries following zero mean and variance
$$
G_{ij} \sim \frac{1}{\sigma_{i,j}*T(T-n-1))}
$$ from the paper [1] below.
I am curious about a case where the entries of $X$, $X_{ij}\sim \mathcal{N}(\mu_{i,j},\sigma_{i,j})$. What would be the distribution of the general inverse $G^{-1}$ for such non-zero-centered data matrix?
Reference
[1] R. DennisCook and Liliana Forzani, "On the mean and variance of the generalized inverse of a singular Wishart matrix" (English) Electronic Journal of Statistics 5, 146-158 (2011), MR2786485, Zbl 1274.62350.
