This review covers recent results concerning the estimation of large
covariance matrices using tools from Random Matrix Theory (RMT). We introduce
several RMT methods and analytical techniques, such as the Replica formalism
and Free Probability, with an emphasis on the Marchenko-Pastur equation that
provides information on the resolvent of multiplicatively corrupted noisy
matrices. Special care is devoted to the statistics of the eigenvectors of the
empirical correlation matrix, which turn out to be crucial for many
applications. We show in particular how these results can be used to build
consistent "Rotationally Invariant" estimators (RIE) for large correlation
matrices when there is no prior on the structure of the underlying process. The
last part of this review is dedicated to some real-world applications within
financial markets as a case in point. We establish empirically the efficacy of
the RIE framework, which is found to be superior in this case to all previously
proposed methods. The case of additively (rather than multiplicatively)
corrupted noisy matrices is also dealt with in a special Appendix. Several open
problems and interesting technical developments are discussed throughout the
paper.

该论文回顾了最近利用随机矩阵理论（RMT）工具估计大协方差矩阵的结果，介绍了几种 RMT 方法和分析技术，如复制品形式和免费概率等，并强调了 Marchenko-Pastur 方程，它提供了有关成倍污染的嘈杂矩阵的解的信息，特别关注经验相关矩阵的特征向量的统计学，说明这些结果特别适用于当没有先验关于基础过程结构的情况下构建大协方差矩阵的一致的 “旋转不变” 估算器（RIE），最后将一些真实世界的应用作为典型案例，确立 RIE 框架的实证效力，在这种情况下，发现其优于以前提出的所有方法。