径向Marchenko-Pastur律：投影刻画与刚性

中文摘要 AI 辅助

设\\(R_p=\\|x_p\\|^2/p\Rightarrow \nu\\)且\\(p/N\to c\in(0,\infty)\\)。我们证明径向二次型条件（RQC），即要求确定性子空间中的能量遵循父半径分布，等价于以下谱性质：对于某个固定的\\(\alpha\in(0,1)\\)，每个确定性秩为\\(\lfloor \alpha p\rfloor\\)的投影协方差矩阵的经验谱分布收敛到\\(\mu_{c\alpha,\nu}\\)。半径律\\(\nu\\)可以具有任意尾部；不需要矩假设、条件各向同性，也不要求半径与方向之间的独立性。我们还证明，当投影谱极限的共同极限具有有限二阶矩且相对维度趋于零的子空间携带的归一化能量趋于零时，可以从投影谱极限恢复半径律，同样不需要对原始向量作矩假设。一个有限样本逆估计通过期望投影谱的有界检验来界定二次型缺陷。逆向论证将角度对称化与fantope刚性相结合，并利用径向列删除与谱修剪之间的行列式比较。删除的径向因子的抵消使得比较误差仅依赖于删除比例，而非删除半径的大小。

英文摘要

Let \(R_p=\|x_p\|^2/p\Rightarrow ν\) and \(p/N\to c\in(0,\infty)\). We prove that the radial quadratic-form condition (RQC), requiring energy in deterministic subspaces to follow the parent radius, is equivalent to the following spectral property: for one fixed \(α\in(0,1)\), every deterministic rank-\(\lfloor αp\rfloor\) projected covariance has empirical spectral distribution converging to \(μ_{cα,ν}\). The radius law \(ν\) may have arbitrary tails; no moment assumptions, conditional isotropy, or independence between radius and direction are required. We also show that the radius law can be recovered from the projected spectral limits when their common limit has finite second moment and subspaces of vanishing relative dimension carry vanishing normalized energy, again without moment assumptions on the original vectors. A finite-sample inverse estimate bounds quadratic-form defects by bounded tests of expected projected spectra. The converse argument combines angular symmetrization and fantope rigidity with a determinant comparison between radial column deletion and spectral trimming. A cancellation of the deleted radial factors makes the comparison error depend only on the deleted fraction, rather than on the magnitudes of the deleted radii.

发表机构

• University of California, Irvine（加州大学尔湾分校）

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