嵌套复样本协方差矩阵的分数对数定律
A Law of Fractional Logarithm for Nested Complex Sample Covariance Matrices
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中文总结 AI 辅助
本文证明嵌套复样本协方差矩阵最大特征值的分数对数定律，给出波动端点与簇集，与Wigner-minor定律一致，采用高斯Laguerre估计和条件比较方法。
中文摘要 AI 辅助
我证明了沿西北嵌套路径的复样本协方差矩阵最大特征值的几乎必然分数对数定律及相应的有限尾簇集，这些矩阵来自一个独立同分布条目的无限阵列。条目具有零均值、单位方差、消失的复二阶矩以及每个固定阶数的均匀有界矩，但不假设为高斯分布；行维度遵循非递减有界增量路径，且具有正极限纵横比。在有限$N$软边居中与缩放后，上下波动端点分别为$(\frac{1}{4})^{2/3}$（在$(\frac{\log N})^{2/3}$尺度上）和$-4^{1/3}$（在$(\frac{\log N})^{1/3}$尺度上），相应的有限尾簇集为$[0,(\frac{1}{4})^{2/3}]$和$[-4^{1/3},\infty)$。这些数值和集合几何与已确立的$\beta=2$ Wigner-minor定律一致。协方差设置需要单独论证，因为行和列同时增长，矩阵共享一个揭示的西北过去，且矩形线性化有多个方向，包括具有确定性零模式的宽情形。我使用高斯Laguerre中等偏差估计和伴随论文的分离网格矩结果。然后通过保持过去固定的条件比较转移所得的发生界，将添加的行和列条带连接到高斯参考族，并在公共过滤上应用递推。确定性无向下跳跃论证将端点极限提升到完整簇集。
英文摘要
We prove a law of fractional logarithm for the largest eigenvalue along a northwest-nested path of complex sample covariance matrices from one infinite array. The entries are independent and centered, with unit variance, vanishing complex second moment, and uniformly bounded moments of every fixed order. The row dimension is nondecreasing, has bounded increments, and has a positive limiting aspect ratio. After finite-size edge centering and scaling, the almost-sure limsup on the $(\log N)^{2/3}$ scale is $(1/4)^{2/3}$, and the liminf on the $(\log N)^{1/3}$ scale is $-4^{1/3}$. The corresponding cluster sets in $\mathbb{R}$ are $[0,(1/4)^{2/3}]$ and $[-4^{1/3},\infty)$. The proof compares the Laplace transform of a single smoothed count over a growing grid of full nested matrices with its Gaussian counterpart. Gaussian count concentration gives block occurrences with probability tending to one. Dyadic tail bounds yield the endpoints, and deterministic interpolation gives the cluster sets.
发表机构
- School of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院）
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