AI 中文总结
该研究针对高维样本相关矩阵平方的迹，在不同正则变化指数α下分析其波动的高斯性，推导了普适与非普适中心极限定理，并通过模拟验证结果。
AI 中文摘要
我们针对由p维随机向量的n个观测值构造的样本相关矩阵平方的迹tr(R²)提供极限理论，其中随机向量的分量独立同分布。若分量具有有限四阶矩且p与n成比例增长，已知tr(R²)满足中心极限定理（CLT），且中心化与缩放序列具有普适性，不依赖于分量分布。在具有指数α的对称性和正则变化假设下，无论维度增长速率如何，我们证明当α>3时该普适CLT仍然成立；当α<3时，我们确定了一个临界维度增长值，超过该值后tr(R²)的波动变为非高斯；此外，若维度p增长更快且α≤3，我们建立了依赖于α值的非普适CLT，其归一化序列与α相关，研究结果通过模拟研究得到验证。
英文摘要
We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfies a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under symmetry and regular variation assumption with index $α$ and any growth rate of the dimension, we prove that the universal CLT remains valid for $α>3$. For $α<3$, we identify a critical dimension growth at which the fluctuations of $\operatorname{tr}({\mathbf R}^2)$ become non-Gaussian. Moreover, if the dimension $p$ grows faster and $α\le 3$ we establish a non-universal CLT with norming sequences depending on the value of $α$. Our findings are illustrated in a simulation study.
Comments50 pages
