Various statistical problems can be formulated in terms of a bilinear form of the covariance matrix. Examples are testing whether coordinates of a high-dimensional random vector are uncorrelated, constructing confidence intervals for the risk of optimal portfolios or testing for the stability of a covariance matrix, especially for factor models. Extending previous works to a general high-dimensional multivariate linear process framework and factor models, we establish distributional approximations for the associated bilinear form of the sample covariance matrix. These approximations hold for increasing dimension without any constraint relative to the sample size. The results are used to construct change-point tests for the covariance structure, especially in order to check the stability of a high-dimensional factor model. Tests based on the cumulated sum (CUSUM), self-standardized CUSUM and the CUSUM statistic maximized over all subsamples are considered. Size and power of the proposed testing methodology are investigated by a simulation study and illustrated by analyzing the Fama and French factors for a change due to the SARS-CoV-2 pandemic.

中文翻译：

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多元线性时间序列和因子模型的高维协方差矩阵的大样本近似和变化检验

可以根据协方差矩阵的双线性形式来表述各种统计问题。示例是测试高维随机向量的坐标是否不相关，构建最佳投资组合风险的置信区间或测试协方差矩阵的稳定性，尤其是对于因子模型。将以前的工作扩展到一般的高维多元线性过程框架和因子模型，我们为样本协方差矩阵的相关双线性形式建立分布近似。这些近似值适用于增加维度，而不受样本大小的任何限制。结果用于构建协方差结构的变点检验，特别是为了检查高维因子模型的稳定性。考虑了基于累积总和 (CUSUM)、自标准化 CUSUM 和在所有子样本上最大化的 CUSUM 统计量的测试。通过模拟研究调查了所提出的测试方法的规模和效力，并通过分析 Fama 和 French 因素对 SARS-CoV-2 大流行引起的变化进行了说明。