Suppose $X$ is random matrix with rows ${X_1^T, X_2^T, \ldots X_N^T}$ where each of the $X_i$ is an independent random vector with the Gaussian distribution $\mathcal{N}(0,K)$ ($K$ being the covariance matrix).
Can we say something about the probabilistic bounds of extremum singular values of $X$ ($\sigma_{min}(X)$ and $\sigma_{max}(X)$) by extending an existing theorem stated as below:
Let $A$ be an $N \times n$ matrix whose rows $A_i$ are independent sub-gaussian isotropic random vectors in $\mathbb{R}^n$. Then for every $t \geq 0$, with probability at least ($1-2\exp(-ct^2))$ one has \begin{equation} \sqrt{N} - c\sqrt{N} - t \leq \sigma_{min}(A) \leq \sigma_{max}(A) \leq \sqrt{N} + c\sqrt{N} + t \end{equation}