大Gram矩阵谱波动的相变：具有方差分布的稀疏CLTs统一框架
Phase Transition of Spectral Fluctuations in Large Gram Matrices with a Variance Profile: A Unified Framework for Sparse CLTs
AI总结：
本文研究了大Gram矩阵在不同稀疏性下的谱波动相变，揭示了高稀疏性下四阶矩主导的波动行为及修正CLT的必要性。
摘要：
我们研究了具有稀疏性和方差分布的高维随机Gram矩阵的渐近谱行为，受无线通信应用的启发。具体来说，我们考虑Gram矩阵S_n=Y_nY_n^*，其中Y_n的元素是独立的、中心化的、异方差的并通过伯努利掩码稀疏化。稀疏性水平参数化为s=q²/n，其中q的范围从多项式阶到n^{1/2}阶。我们研究了两种渐近情形：稀疏性固定s∈(0,1]的中等稀疏性情形，以及s→0的高稀疏性情形。在两种情形中，我们建立了S_n的经验谱分布收敛到确定性极限，并进一步利用解析技术与鞅差论证推导线性谱统计量的中心极限定理。我们的分析揭示了两种情形间波动行为的相变。在高稀疏性情形中，渐近波动完全由四阶矩效应决定，稀疏性缩放贡献被抑制。此外，主导的确定性项和线性谱统计量的方差在q中的缩放速率不同，导致标准中心化失效，必须通过显式修正来恢复有效的CLT。结果适用于高斯和非高斯元素，并通过应用于大规模MIMO系统中的假设检验和 outage 概率分析来展示。
英文摘要：
We study the asymptotic spectral behavior of high-dimensional random Gram matrices with sparsity and a variance profile, motivated by applications in wireless communications. Specifically, we consider the Gram matrices $\mathbf S_n=\mathbf Y_n\mathbf Y_n^*$, where the entries of $\mathbf Y_n$ are independent, centered, heteroscedastic, and sparse through Bernoulli masking. The sparsity level is parameterized as $s=q^2/n$, where $q$ ranges from polynomial order up to order $n^{1/2}$. We investigate two asymptotic regimes: a moderate-sparsity regime with fixed $s\in(0,1]$, and a high-sparsity regime where $s\to0$. In both regimes, we establish the convergence of the empirical spectral distribution of $\mathbf S_n$ to a deterministic limit, and further derive central limit theorems for linear spectral statistics using resolvent techniques and martingale difference arguments. Our analysis reveals a phase transition in the fluctuation behavior across the two regimes. In the high-sparsity regime, the asymptotic fluctuations are entirely governed by fourth-moment effects, with sparsity-scaled contributions being suppressed. Moreover, the leading deterministic term and the variance of the linear spectral statistic scale at different rates in $q$, causing the standard centering to fail and necessitating an explicit correction to recover a valid CLT. The results apply to both Gaussian and non-Gaussian entries and are illustrated through applications to hypothesis testing and outage probability analysis in large-scale MIMO systems.
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