This paper concerns about the limiting distributions of change point estimators, in a high-dimensional linear regression time series context, where a regression object $(y_t, X_t) \in \mathbb{R} \times \mathbb{R}^p$ is observed at every time point $t \in \{1, \ldots, n\}$. At unknown time points, called change points, the regression coefficients change, with the jump sizes measured in $\ell_2$-norm. We provide limiting distributions of the change point estimators in the regimes where the minimal jump size vanishes and where it remains a constant. We allow for both the covariate and noise sequences to be temporally dependent, in the functional dependence framework, which is the first time seen in the change point inference literature. We show that a block-type long-run variance estimator is consistent under the functional dependence, which facilitates the practical implementation of our derived limiting distributions. We also present a few important byproducts of our analysis, which are of their own interest. These include a novel variant of the dynamic programming algorithm to boost the computational efficiency, consistent change point localisation rates under temporal dependence and a new Bernstein inequality for data possessing functional dependence. Extensive numerical results are provided to support our theoretical results. The proposed methods are implemented in the R package \texttt{changepoints} \citep{changepoints_R}.

翻译：本文关注高维线性回归时间序列背景下变点估计量的极限分布，其中在每个时间点 $t \in \{1, \ldots, n\}$ 观测到回归对象 $(y_t, X_t) \in \mathbb{R} \times \mathbb{R}^p$。在未知时间点（称为变点）处，回归系数发生变化，跳跃大小以 $\ell_2$ 范数度量。我们提供了最小跳跃大小趋于零及保持常数两种情形下变点估计量的极限分布。我们允许协变量和噪声序列在函数依赖框架下具有时间依赖性，这在变点推断文献中尚属首次。我们证明在函数依赖下块型长期方差估计量具有一致性，这有助于我们推导出的极限分布的实际应用。此外，我们提出了若干具有独立意义的重要分析副产品，包括一种提升计算效率的动态规划算法新变体、时间依赖下一致的变点定位率，以及针对具有函数依赖数据的新 Bernstein 不等式。广泛的数值结果支持了我们的理论结果。所提出的方法已在 R 包 \texttt{changepoints} \citep{changepoints_R} 中实现。