独立同分布次高斯随机矩阵谱半径的大偏差

中文摘要 AI 辅助

我们研究 $X_n=n^{-1/2}(\xi_{ij})_{i,j=1}^n$ 的谱半径的大偏差，其中条目是独立同分布、中心化、方差为1且次高斯的，在复情形下伪方差为零。对每个固定的 $r>1$，我们证明 $$ \liminf_{n\to\infty}\frac1n\log\mathbb{P}\{\rho(X_n)>r\} \ge -\frac\beta2\bigl(r^2-1-2\log r\bigr), $$ 其中实条目时 $\beta=1$，复条目时 $\beta=2$。我们为具有高斯主导偶矩的实对称分布以及满足尖锐高斯拉普拉斯变换界的复分布证明了匹配的上界。这些类别包含离散分布。在复尖锐类中，我们还确定了特征值进入单位圆盘外固定圆盘的指数速率。然后我们转向下偏差并建立二次速度界。在额外假设条目分布 $\mu$ 具有有界密度的条件下，我们证明对每个固定的 $0<r<1$ 和所有足够大的 $n$，有 $$ \mathbb{P}\{\rho(X_n)\le r\}\le e^{-c_{\mu,r}n^2} $$，其中 $c_{\mu,r}>0$。对于具有有界密度的匹配上尾类，这些界给出了完整的 $n$ 速度大偏差原理，其中 $r\ge1$ 时采用所示的上尾速率，$r<1$ 时速率为无穷。上尾证明使用了一种保持条目支撑并产生离群特征值的测度变换。下尾证明为自适应正交观测开发了加权比较，并将其应用于Arnoldi残差。

英文摘要

We study large deviations of the spectral radius of $X_n=n^{-1/2}(ξ_{ij})_{i,j=1}^n$, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. For every fixed $r>1$, we prove $$ \liminf_{n\to\infty}\frac1n\log\mathbb{P}\{ρ(X_n)>r\} \ge -\frac\beta2\bigl(r^2-1-2\log r\bigr), $$ where $β=1$ for real entries and $β=2$ for complex entries. We prove a matching upper bound for real symmetric laws with Gaussian-dominated even moments and for complex laws satisfying the sharp Gaussian Laplace-transform bound. These classes include discrete distributions. In the complex sharp class, we also identify the exponential rate for an eigenvalue to enter a fixed disk outside the unit disk. We then turn to lower deviations and establish a quadratic-speed bound. Under the additional assumption that the entry law $μ$ has a bounded density, we prove $$ \mathbb{P}\{ρ(X_n)\le r\}\le e^{-c_{μ,r}n^2} $$ for every fixed $0<r<1$ and all sufficiently large $n$, where $c_{μ,r}>0$. For the matching upper-tail classes with a bounded density, these bounds give a full speed-$n$ large deviation principle, with the displayed upper-tail rate for $r\ge1$ and infinite rate for $r<1$. The upper-tail proof uses a change of measure that preserves the entry support and creates an outlying eigenvalue. The lower-tail proof develops a weighted comparison for adaptive orthonormal observations and applies it to Arnoldi residuals.

发表机构

• Massachusetts Institute of Technology（麻省理工学院）

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77 pages