Spectral Moments of Random Matrices with a Rank-One Pattern of Variances

Abstract

Let ${\mathbf{a}}_{ij}$, $1\le i\le j\le n$, be independent random variables and ${\mathbf{a}}_{ji}={\mathbf{a}}_{ij}$, for all $i,j$. Suppose that every ${\mathbf{a}}_{ij}$ is bounded, has zero mean, and its variance is given by ${\sigma }_i{\sigma }_j$, for a given sequence of positive real numbers $\Psi =\{{\sigma }_i,i\in \mathbb{N}\}$. Hence, the matrix of variances $V_n=\left( \mbox{Var}( \mathbf{a}_{ij}) \right)_{i,j=1}^n$ has rank one for all $n$. We show that the empirical spectral distribution of the symmetric random matrix ${\mathbf{A}}_n(\Psi )$ $=$ ${\left({\mathbf{a}}_{ij}/\sqrt{n}\right)}_{i,j=1}^n$ converges weakly (and with probability one) to a deterministic limiting spectral distribution which we fully characterize by providing closed-form expressions for its limiting spectral moments in terms of the sequence $\Psi$. Furthermore, we propose a hierarchy of semidefinite programs to compute upper and lower bound on the expected spectral norm of ${\mathbf{A}}_n$, for both finite $n$ and the limit $n\to \infty$.

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