嵌套复Wishart极值的单侧乘积尺度上界
One-Sided Product-Scale Upper Bounds for Nested Complex Wishart Extremes
AI总结：
本文针对嵌套复Wishart极值的最大特征值软边缘，证明了其上、下尾联合概率的有限N单侧乘积尺度上界，推导了分离稀疏网格的矩估计，适用于超临界分离场景。
AI中文摘要：
本文研究无限复高斯阵列的第Mj×j西北矩形块X^{(j)}的最大特征值Λ_j，即X^{(j)}(X^{(j)})^*的最大特征值。针对最大特征值软边缘处的两个上尾事件和两个下尾事件，本文证明了它们联合概率的有限N单侧乘积尺度上界。对数墙具有多项式小的边缘概率，因此加性o(1)协方差估计可能远大于需控制的消失乘积。在短宏观窗口内的水平、有界确定性偏移以及至少N^{2/3+ε}的分离度下，每个联合概率至多为其边缘概率乘积的(1+o(1))倍。经典的N^{2/3}相关窗口对应一阶扩展Airy时间；该定理适用于超临界分离。沿容许行-列路径的精确Laguerre算子提供有限维输入。上尾证明使用占有计数和跨块迹估计，下尾证明使用间隙行列式、对角预解式和Schur补。本文在半整数Laguerre与物理矩形归一化之间转移结果，并针对稀有性、计数增长、间距和宏观窗口预算，推导分离稀疏网格的固有一阶和二阶矩估计。
英文摘要：
I study the largest eigenvalue \(Λ_j\) of \(X^{(j)}(X^{(j)})^*\), where \(X^{(j)}\) is the \(M_j\times j\) northwest rectangle of one infinite complex Gaussian array. For two upper-tail events and, separately, for two lower-tail events at the largest-eigenvalue soft edge, I prove finite-\(N\) one-sided product-scale upper bounds for their joint probabilities. The logarithmic walls have polynomially small marginals, so an additive \(o(1)\) covariance estimate may be much larger than the vanishing product that must be controlled. Uniformly for levels in a short macroscopic window, bounded deterministic shifts, and separations at least \(N^{2/3+ε}\), each joint probability is at most \((1+o(1))\) times the product of its marginals. The classical \(N^{2/3}\) correlation window corresponds to order-one extended-Airy time; the theorem works at supercritical separation. Exact Laguerre operators along admissible row--column paths provide the finite-dimensional input. The upper-tail proof uses occupancy counts and cross-block trace estimates, whereas the lower-tail proof uses gap determinants, diagonal resolvents, and a Schur complement. I transfer the result between the half-integer Laguerre and physical rectangular normalizations and deduce intrinsic first- and second-moment estimates for separated sparse grids under the rarity, count-growth, spacing, and macroscopic-window budget.
