In this article, we consider a $n$-dimensional random walk $X_t$, whose error terms are linear processes generated by $n$-dimensional noise vectors, and each component of these noise vectors is independent and may not be identically distributed with uniformly bounded 8th moment and densities. Given $T$ observations such that the dimension $n$ and sample size $T$ going to infinity proportionally, define $\boldsymbol{X}$ and $\hat{\boldsymbol{R}}$ as the data matrix and the sample correlation matrix of $\boldsymbol{X}$ respectively. This article establishes the central limit theorem (CLT) of the first $K$ largest eigenvalues of $n^{-1}\hat{\boldsymbol{R}}$. Subsequently, we propose a new unit root test for the panel high-dimensional nonstationary time series based on the CLT of the largest eigenvalue of $n^{-1}\hat{\boldsymbol{R}}$. A numerical experiment is undertaken to verify the power of our proposed unit root test.

翻译：本文考虑一个$n$维随机游走$X_t$，其误差项由$n$维噪声向量生成的线性过程构成，且这些噪声向量的每个分量独立但未必同分布，且具有一致有界的8阶矩和密度函数。基于$T$个观测值，其中维度$n$与样本量$T$按比例趋于无穷大，定义$\boldsymbol{X}$和$\hat{\boldsymbol{R}}$分别为数据矩阵和$\boldsymbol{X}$的样本相关矩阵。本文建立了$n^{-1}\hat{\boldsymbol{R}}$的前$K$个最大特征值的中心极限定理（CLT）。随后，基于$n^{-1}\hat{\boldsymbol{R}}$最大特征值的CLT，我们提出了一种新的面板高维非平稳时间序列的单位根检验。数值实验验证了所提出的单位根检验的统计功效。