We present two sharp empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by the first-order $1/\sqrt{n}$ term asymptotically matches the matrix Bernstein inequality exactly, including constants, the latter requiring knowledge of the variance. Our first inequality holds for the sample mean of independent matrices, and our second inequality holds for a mean estimator under martingale dependence at stopping times.

翻译：针对具有有界特征值的对称随机矩阵，我们提出了两个锐利的经验伯恩斯坦不等式。所谓锐利，是指两个不等式均以紧致的方式适应未知方差：其一阶$1/\sqrt{n}$项所捕获的偏差在渐近意义上精确匹配矩阵伯恩斯坦不等式（包括常数项），而后者需要已知方差信息。我们的第一个不等式适用于独立矩阵的样本均值，第二个不等式则适用于停时下鞅依赖情形的均值估计量。