AI 中文总结
研究尺度混合数据的高维斯皮尔曼相关矩阵渐近谱性质，通过特定观测形式分析其高阶相依性，在比例情形下得出经验谱分布收敛规律，还给出潜在变量扩展及可解示例与数值近似，受重尾数据相关领域启发。
AI 中文摘要
我们研究了尺度混合数据的高维斯皮尔曼相关矩阵的渐近谱性质。考虑形如\(x_t = \sigma_t \xi_t \in \mathbb{R}^N\)的观测值，其中\(\xi_t\)的坐标独立同分布，标量混合变量\(\sigma_t\)为所有坐标所共有。在自然对称假设下，\(x_t\)的坐标在皮尔逊和斯皮尔曼意义下两两不相关，但当混合变量非退化时它们并非独立。我们表明这种高阶相依在秩变换后依然存在并留下非平凡的谱特征。在比例情形\(N/T \to q \in (0, \infty)\)下，斯皮尔曼相关矩阵的经验谱分布几乎必然收敛到由有效秩方差的极限分布所支配的广义马尔琴科 - 帕斯特尔定律。我们还提出了一个更广泛的潜在变量扩展，特别涵盖了一些具有相关方向分量的尺度混合模型。我们讨论了可解示例和数值近似，部分动机来自稳健多元统计、计量经济学和金融中的重尾数据。
英文摘要
We study the asymptotic spectral properties of high-dimensional Spearman correlation matrices for scale-mixture data. We consider observations of the form $x_t=σ_t ξ_t \in \mathbb{R}^N,$ where the coordinates of $ξ_t$ are i.i.d.\ and the scalar mixture variable $σ_t$ is shared by all coordinates. Under natural symmetry assumptions, the coordinates of $x_t$ are pairwise uncorrelated in both the Pearson and Spearman sense. Nevertheless, they are not independent when the mixture variable is non-degenerate. We show that this higher-order dependence survives the rank transformation and leaves a nontrivial spectral signature. In the proportional regime $N/T\to q\in(0,\infty),$ the empirical spectral distribution of the Spearman correlation matrix converges almost surely to a generalized Marčenko--Pastur law governed by the limiting distribution of an effective rank variance. We also formulate a broader latent-variable extension, which covers, in particular, some scale-mixture models with correlated directional components. We discuss solvable examples and numerical approximations, motivated in part by heavy-tailed data in robust multivariate statistics, econometrics, and finance.
Comments24 pages, 4 figures
