We prove local laws for the resolvents of separable covariance matrices of the form $\mathcal Q=A^{1/2}XBX^*A^{1/2}$, where $X=(x_{ij})$ is a $p\times n$ random matrix whose entries $x_{ij}$ are i.i.d.~random variables with mean 0 and variance $n^{-1}$, and $A,B$ are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of $\mathcal Q$ and use it to prove optimal local laws without the technical assumption $\mathbb{E}[x_{ij}^{3}]=0$, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.
We consider the real noncentral sample covariance matrices $\mathcal{W}=YY^\top$ with $Y=A+\Sigma^{1/2}X$. Here $A\in\mathbb{R}^{M\times N}$ is deterministic, $\Sigma\in \mathbb{R}^{M\times M}$ is a deterministic positive definite population covariance matrix and $X\in\mathbb{R}^{M\times N}$ has independent centered en...
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance,...
We initiate the study of approximating the top eigenvalue and eigenvector of a random symmetric matrix $ A \in \mathbb{R}^{n\times n} $ using $ q(A)b $ where $q$ is a degree-$d$ polynomial and $b$ is a standard Gaussian vector independent of $A$. For spiked GOE $ Y = \lambda vv^\top + X $, we identify $ d_\star = \frac...
We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfie...
We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_...
Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establis...
August Y. Chen, A. El Alaoui· 0 citations
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