AI 中文总结
该研究在总体相关矩阵正定且样本自由度不小于维度的条件下，证明了高斯皮尔逊样本相关矩阵对数行列式的中心极限定理，其几何意义与随机平行六面体对数体积相关。
AI 中文摘要
我们证明了当维度发散时，高斯皮尔逊样本相关矩阵对数行列式的中心极限定理。仅施加两个条件：总体相关矩阵是正定的，且样本自由度至少等于维度。这两个条件对于普通对数行列式有限是必要的。据我们所知，此前没有任何中心极限定理覆盖这个完整的非奇异域，它涵盖了从稀疏增长到方形硬边界的所有长宽比。对总体相关矩阵的特征值未施加任何一致的下界或上界：最小特征值可趋近于零，最大特征值可发散。该证明为随机对角归一化建立了逐坐标的维纳混沌约化，并将其与精确的威沙特变换比较相结合。从几何角度看，该统计量是由标准化高斯坐标向量张成的随机平行六面体对数体积的两倍。
英文摘要
We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.
Comments34 pages, 1 table
