发表机构
Harbin Institute of Technology; Nanyang Technological University; Jinan University; National University of Singapore; Changchun University of Science and Technology（哈尔滨工业大学; 南洋理工大学; 暨南大学; 新加坡国立大学; 长春理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究重尾样本相关矩阵最大特征值的阶梯相变，发现其由长宽比和条目尾部共同决定，临界尾部指数随长宽比阶梯式下降，并给出泊松极限及非退化极限定律。
AI 中文摘要
我们建立了由 $p_n \times n$ 数据矩阵（其元素独立同分布，均值为零，方差为一，且允许四阶矩无穷）形成的重尾样本相关矩阵最大特征值的阶梯相变。在比例极限 $p_n / n \to \phi \in (0, \infty)$ 下，一阶渐近行为同时依赖于长宽比和条目尾部。相变由同一列中不同行的较大条目之间的碰撞驱动。能够产生分离的上离群值的第一个碰撞阶数为 $k_* (\phi) = \lfloor \sqrt{\phi} \rfloor + 2$，由此得到临界尾部指数 $\alpha_* (\phi) = 2 + 2 / k_* (\phi)$。该指数随着 $\phi$ 的增加而阶梯式下降，在收敛到上 Marchenko--Pastur 边缘和连续的离群值水平之间形成阶梯边界。在精确的临界尾部尺度下，上边缘或先前确定性水平之上的特征值点过程收敛到泊松点过程。由此得到的最大特征值的非退化极限定律连接了相邻相，并在该基线上具有正原子。如果每个固定的碰撞阶数都是超临界的，则尽管条目方差有限，最大特征值仍在概率上发散。
英文摘要
We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a $p_n \times n$ data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime $p_n / n \to ϕ\in (0, \infty)$, the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is $k_* (ϕ) = \lfloor \sqrtϕ \rfloor + 2$, yielding the critical tail exponent $α_* (ϕ) = 2 + 2 / k_* (ϕ)$. This exponent decreases in steps as $ϕ$ increases, creating a staircase boundary between convergence to the upper Marčenko--Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.
