We present an estimator of the covariance matrix $\Sigma$ of random $d$-dimensional vector from an i.i.d. sample of size $n$. Our sole assumption is that this vector satisfies a bounded $L^p-L^2$ moment assumption over its one-dimensional marginals, for some $p\geq 4$. Given this, we show that $\Sigma$ can be estimated from the sample with the same high-probability error rates that the sample covariance matrix achieves in the case of Gaussian data. This holds even though we allow for very general distributions that may not have moments of order $>p$. Moreover, our estimator can be made to be optimally robust to adversarial contamination. This result improves recent results in the literature by Mendelson and Zhivotovskiy and Catoni and Giulini, and matches parallel work by Abdalla and Zhivotovskiy (the exact relationship with this last work is described in the paper).

翻译：我们从一.d. 大小样本中提出一个共变矩阵的估算值 $\ sigma$, 随机的美元- 维向量的估算值。 我们唯一的假设是, 该矢量满足了对其一维边际的受约束的 $Lp- L ⁇ 2 秒假设值, 大约为 $p\ geq 4 美元。 有鉴于此, 我们显示, $\ sigma$ 可以从样本中估算出, 与样本共变矩阵在高斯数据中达到的高概率误差率相同。 尽管我们允许非常笼统的分布, 且可能没有 $> p$ 。 此外, 我们的估量器可以优化地适应对抗性污染。 这改善了Mendelson 和 Zhivotovskiy 以及 Catoni 和 Giulini 的文献的最新结果, 与 Abdalla 和 Zhivotovskiy 的平行工作( 与本文中所描述的准确关系) 。