Edge statistics of large dimensional deformed rectangular matrices
Edge statistics of large dimensional deformed rectangular matrices
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DOI：
10.1016/j.jmva.2022.105051
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发表时间：
2022-06-11
影响因子：
1.6
通讯作者：
Yang，Fan
中科院分区：
文献类型：
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作者：
Ding，Xiucai;Yang，Fan
我们考虑了形式为Yt=Y+tX的大维变形矩形矩阵的边缘统计量，其中Y是一个p×n的确定性信号矩阵，其秩可比于n，X是一个具有均值零和方差n−1的iid项的p×n随机噪声矩阵，t>；0给出了噪声水平。在大规模多输入多输出(MIMO)系统的研究中，该模型被称为干扰加噪声矩阵，属于信号加噪声模型的范畴。对于t=1的情况，文献中已经在一定程度上研究了该模型的谱统计(Dozier和Silverstein，2007[17，18]；Vallet等人，2012)。本文研究了在较硬区域n−1/3≪t≪1中，Y t在谱的最右边缘附近的奇异值和奇异向量统计量。这一区域比t=1的情况更困难，因为一方面，Y Y⊤的经验谱分布的边缘行为对Y t Y t⊤的边缘统计有很大的影响，而另一方面，对于较小的t≪1，Y t Y t⊤的边本征值行为不仅仅是Y Y⊤对于“大”t≫n−1/3的边本征值行为的扰动.在Y上的某些正则性假设下，我们证明了Y t Y t⊤和Y t⊤Yt的边本征值的特雷西-维多姆定律，以及特征向量的离域化.这些结果可用于大规模多输入多输出系统的估计和推断.为了证明主要结果，我们分析了Y-t-Y-t-⊤的渐近esd的边缘行为，并在其预解上建立了最优局部律。这些结果是独立感兴趣的，并且可以作为其他许多关于Yt的谱统计问题的重要输入。
We consider the edge statistics of large dimensional deformed rectangular matrices of the form Y t= Y+ t X, where Y is a p× n deterministic signal matrix whose rank is comparable to n, X is a p× n random noise matrix with iid entries of mean zero and variance n− 1, and t> 0 gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case t= 1, the spectral statistics of this model have been studied to a certain extent in the literature (Dozier and Silverstein, 2007 [17, 18]; Vallet et al., 2012). In this paper, we study the singular value and singular vector statistics of Y t around the right-most edge of the spectrum in the harder regime n− 1/3≪ t≪ 1. This regime is harder than the t= 1 case, because on the one hand, the edge behavior of the empirical spectral distribution (ESD) of Y Y⊤ has a strong effect on the edge statistics of Y t Y t⊤ for a “small” t≪ 1, while on the other hand, the edge eigenvalue behavior of Y t Y t⊤ is not merely a perturbation of that of Y Y⊤ for a “large” t≫ n− 1/3. Under certain regularity assumptions on Y, we prove the Tracy–Widom law for the edge eigenvalues, the eigenvalue rigidity, and eigenvector delocalization for the matrices Y t Y t⊤ and Y t⊤ Y t. These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD of Y t Y t⊤ and establish optimal local laws on its resolvent. These results are of independent interest, and can be used as important inputs for many other problems regarding the spectral statistics of Y t.