Let $X\in\mathbb{R}^{n\times d}$ be a random matrix with independent rows which satisfy that $\mathbb{E}(X_iX_i^T) = I_d$ and $\mathbb{E}(X_i) = 0$. The results of P. Yaskov provide conditions on when the minimum eigenvalue of the covariance matrix $X^TX/n\in\mathbb{R}^{d\times d}$ is lower bounded in the case that $d/n \to \rho <1$.
Is it possible to obtain conditions on when the smallest eigenvalue of $XX^T/n\in\mathbb{R}^{n\times n}$ (so the smallest positive eigenvalue of $X^TX/n$) is lower bounded in the case that $d/n \to \rho \geq 1$.