AI 中文总结
该研究推导了样本相关矩阵间对数欧氏距离的渐近抽样分布，给出多种尾概率校准方法与估计误差经验法则，为量化金融等领域的相关矩阵结构变化检验提供理论支撑。
AI 中文摘要
在量化金融和多元统计中，跨时间或压力情景比较相关矩阵至关重要，但样本估计噪声往往会掩盖观测到的距离是否反映了真实的结构变化。我们推导了在总体相关矩阵一致的原假设下，两个独立估计的满秩相关矩阵之间固有对数偏移（对数欧氏）距离的渐近抽样分布。在具有有限四阶矩的一般抽样下，缩放后的平方距离收敛于独立$\chi_1^2$变量的加权和，权重由广义费希尔变换（GFT）坐标的渐近协方差决定。在独立假设下的高斯抽样中，该分布简化为无参数的$4\chi_d^2$分布。为校准尾概率，我们提供了闭式累积量生成函数、Lugannani–Rice鞍点分位数，以及无需寻根的显式Chernoff包络。极限分布的一阶矩为原假设下的基线期望距离建立了一个简单的经验法则（独立附近$\operatorname E[d_{\mathrm{LE}}] \lesssim 2\sqrt{d/n}$），量化了严格由估计误差引起的平均分离度。我们建立了插入一致性，提出了显式高斯协方差分解，将该距离统计量与坐标Wald检验进行了比较，并刻画了其局部势。
英文摘要
Comparing correlation matrices across time or stress scenarios is critical in quantitative finance and multivariate statistics, yet sample estimation noise often obscures whether an observed distance reflects a true structural shift. We derive the asymptotic sampling distribution of the intrinsic off-log (log-Euclidean) distance between two independently estimated full-rank correlation matrices under the null hypothesis that their population correlation matrices coincide. Under general sampling with finite fourth moments, the scaled squared distance converges to a weighted sum of independent $χ_1^2$ variables, with weights determined by the asymptotic covariance of the Generalized Fisher Transformation (GFT) coordinates. Under Gaussian sampling at independence, this simplifies to a parameter-free $4χ_d^2$ law. To calibrate tail probabilities, we provide closed-form cumulant generating functions, Lugannani--Rice saddlepoint quantiles, and an explicit Chernoff envelope requiring no root-finding. The first moment of the limiting law establishes a simple rule of thumb for the baseline expected distance under the null hypothesis ($\operatorname E[d_{\mathrm{LE}}] \lesssim 2\sqrt{d/n}$ near independence), quantifying the average separation induced strictly by estimation error. We establish plug-in consistency, present an explicit Gaussian covariance factorization, compare the distance statistic with coordinate Wald tests, and characterize its local power.
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