Change point detection in covariance structures is a fundamental and crucial problem for sequential data. Under the high-dimensional setting, most of the existing research has focused on identifying change points in historical data. However, there is a significant lack of studies on the practically relevant online change point problem, which means promptly detecting change points as they occur. In this paper, applying the limiting theory of linear spectral statistics for random matrices, we propose a class of spectrum based CUSUM-type statistic. We first construct a martingale from the difference of linear spectral statistics of sequential sample Fisher matrices, which converges to a Brownian motion. Our CUSUM-type statistic is then defined as the maximum of a variant of this process. Finally, we develop our detection procedure based on the invariance principle. Simulation results show that our detection method is highly sensitive to the occurrence of change point and is able to identify it shortly after they arise, outperforming the existing approaches.

翻译：协方差结构中的变点检测是序列数据分析中一个基础且关键的问题。在高维设定下，现有研究大多集中于识别历史数据中的变点。然而，对于实际应用中极为相关的在线变点检测问题——即当变点发生时能够及时检测——目前的研究显著不足。本文应用随机矩阵线性谱统计量的极限理论，提出了一类基于谱的CUSUM型统计量。我们首先从序列样本Fisher矩阵的线性谱统计量之差构造一个鞅，该鞅收敛于一个布朗运动。随后，我们将CUSUM型统计量定义为该过程一个变体的最大值。最后，基于不变原理，我们构建了相应的检测流程。仿真结果表明，我们的检测方法对变点的发生具有高度敏感性，能够在变点出现后迅速识别，其性能优于现有方法。