Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ and $\lambda_{\max}(\sum_{i=1}^n a_ia_i^T)$. What are the best concentration inequalities to use?

Matrix Chernoff bound gives the bound for bounded random matrices. Is the best way here to derive a truncated variant of Matrix Chernoff bound?

Thanks!

Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ and $\lambda_{\max}(\sum_{i=1}^n a_ia_i^T)$. What are the best concentration inequalities to use?

Matrix Chernoff bound gives the bound for bounded random matrices. Is the best way here to derive a truncated variant of Matrix Chernoff bound?

Update: I find the Matrix Bernstein inequality for subexponential matrices (Theorem 6.2). Applying Remark 3.10, we can obtain a bound for the smallest eigenvalue.