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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.04302 |
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| _version_ | 1866910016237207552 |
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| author | Wang, Rui Pan, Guangming Jiang, Dandan |
| author_facet | Wang, Rui Pan, Guangming Jiang, Dandan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Phase Transition of Spectral Fluctuations in Large Gram Matrices with a Variance Profile: A Unified Framework for Sparse CLTs Wang, Rui Pan, Guangming Jiang, Dandan Statistics Theory 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. |
| title | Phase Transition of Spectral Fluctuations in Large Gram Matrices with a Variance Profile: A Unified Framework for Sparse CLTs |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2602.04302 |