In this paper, the key objects of interest are the sequential covariance matrices $\mathbf{S}_{n,t}$ and their largest eigenvalues. Here, the matrix $\mathbf{S}_{n,t}$ is computed as the empirical covariance associated with observations $\{\mathbf{x}_1,\ldots,\mathbf{x}_{ \lfloor nt \rfloor } \}$, for $t\in [0,1]$. The observations $\mathbf{x}_1,\ldots,\mathbf{x}_n$ are assumed to be i.i.d. $p$-dimensional vectors with zero mean, and a covariance matrix that is a fixed-rank perturbation of the identity matrix. Treating $\{ \mathbf{S}_{n,t}\}_{t \in [0,1]}$ as a matrix-valued stochastic process indexed by $t$, we study the behavior of the largest eigenvalues of $\mathbf{S}_{n,t}$, as $t$ varies, with $n$ and $p$ increasing simultaneously, so that $p/n \to y \in (0,1)$. As a key contribution of this work, we establish the weak convergence of the stochastic process corresponding to the sample spiked eigenvalues, if their population counterparts exceed the critical phase-transition threshold. Our analysis of the limiting process is fully comprehensive revealing, in general, non-Gaussian limiting processes. As an application, we consider a class of change-point problems, where the interest is in detecting structural breaks in the covariance caused by a change in magnitude of the spiked eigenvalues. For this purpose, we propose two different maximal statistics corresponding to centered spiked eigenvalues of the sequential covariances. We show the existence of limiting null distributions for these statistics, and prove consistency of the test under fixed alternatives. Moreover, we compare the behavior of the proposed tests through a simulation study.

翻译：本文关注的关键对象是序贯协方差矩阵 $\mathbf{S}_{n,t}$ 及其最大特征值。其中，对于 $t\in [0,1]$，矩阵 $\mathbf{S}_{n,t}$ 由观测值 $\{\mathbf{x}_1,\ldots,\mathbf{x}_{ \lfloor nt \rfloor } \}$ 计算得到。假设观测值 $\mathbf{x}_1,\ldots,\mathbf{x}_n$ 是独立同分布的 $p$ 维零均值向量，其协方差矩阵为单位矩阵的固定秩扰动。我们将 $\{ \mathbf{S}_{n,t}\}_{t \in [0,1]}$ 视为以 $t$ 为索引的矩阵值随机过程，研究当 $n$ 和 $p$ 同步增长（满足 $p/n \to y \in (0,1)$）时，$\mathbf{S}_{n,t}$ 的最大特征值随 $t$ 变化的行为。作为本文的核心贡献，当样本尖峰特征值对应的总体尖峰特征值超过临界相变阈值时，我们建立了其随机过程的弱收敛性。对极限过程的分析全面揭示了非高斯极限过程（一般情况下）。作为应用，我们考虑一类变点问题，其关注点在于检测由尖峰特征值幅度变化引起的协方差结构断点。为此，我们提出了两种基于序贯协方差中心化尖峰特征值的极大统计量。我们证明了这些统计量在零假设下的极限分布存在性，并验证了固定备择假设下检验的一致性。此外，我们通过模拟研究比较了所提检验的行为。