This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix has greater spectrum under this ordering, its estimator will have worse tail bounds. This is done for two families of matrices: positive semidefinite matrices with bounded effective rank, and indefinite matrices with bounded 2-norm and fixed Frobenius norm. In each case, the tail region is defined rigorously and is constant for a given family.

翻译：本文推导了应用于实对称矩阵的高斯迹估计量的极值尾界。我们定义了一个特征值上的偏序关系，使得当矩阵在此偏序下具有更大的谱时，其估计量的尾界将更差。这项工作针对两类矩阵进行：具有有界有效秩的半正定矩阵，以及具有有界2-范数和固定Frobenius范数的不定矩阵。在每种情况下，尾区域均被严格定义，并且对于给定矩阵族是常数。