We study properties of a sample covariance estimate $\widehat \Sigma$ given a finite sample of $n$ i.i.d. centered random elements in $\R^d$ with the covariance matrix $\Sigma$. We derive dimension-free bounds on the squared Frobenius norm of $(\widehat\Sigma - \Sigma)$ under reasonable assumptions. For instance, we show that $\smash{\|\widehat\Sigma - \Sigma\|_{\rm F}^2}$ differs from its expectation by at most $\smash{\mathcal O({\rm{Tr}}(\Sigma^2) / n)}$ with overwhelming probability, which is a significant improvement over the existing results. This allows us to establish the concentration phenomenon for the squared Frobenius distance between the covariance and its empirical counterpart in the case of moderately large effective rank of $\Sigma$.

翻译：我们研究给定$n$个独立同分布中心化随机元素（取值于$\R^d$空间且具有协方差矩阵$\Sigma$）的有限样本时，样本协方差估计量$\widehat \Sigma$的性质。在合理假设下，我们推导出$(\widehat\Sigma - \Sigma)$的Frobenius范数平方的无维界。例如，我们证明$\smash{\|\widehat\Sigma - \Sigma\|_{\rm F}^2}$以压倒性概率与其期望值的差异不超过$\smash{\mathcal O({\rm{Tr}}(\Sigma^2) / n)}$，这相较于现有结果有显著改进。这使得我们能够在$\Sigma$具有适度较大有效秩的情况下，建立协方差矩阵与其经验对应量之间Frobenius距离平方的集中现象。